It is shown in Yoshiara (2004) that, if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16302230&_mathId=si1.gif&_user=111111111&_pii=S0012365X16302230&_rdoc=1&_issn=0012365X&md5=687fe321ae4845dff9bb032121e41e88" title="Click to view the MathML source">dclass="mathContainer hidden">class="mathCode">-dimensional dual hyperovals exist in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16302230&_mathId=si2.gif&_user=111111111&_pii=S0012365X16302230&_rdoc=1&_issn=0012365X&md5=0a29779ee1747cd881f97a29f97ddfaf" title="Click to view the MathML source">V(n,2)class="mathContainer hidden">class="mathCode"> (class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16302230&_mathId=si3.gif&_user=111111111&_pii=S0012365X16302230&_rdoc=1&_issn=0012365X&md5=b10b0dc58a4898bafb619f37e4e3d0f3" title="Click to view the MathML source">GF(2)class="mathContainer hidden">class="mathCode">-vector space of rank class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16302230&_mathId=si4.gif&_user=111111111&_pii=S0012365X16302230&_rdoc=1&_issn=0012365X&md5=ab1d1dffc44cdf1e158e52ed78437bec" title="Click to view the MathML source">nclass="mathContainer hidden">class="mathCode">), then class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16302230&_mathId=si5.gif&_user=111111111&_pii=S0012365X16302230&_rdoc=1&_issn=0012365X&md5=4be7c5f9c6b29bcaab75975a001ebfb7" title="Click to view the MathML source">2d+1≤n≤(d+1)(d+2)/2+2class="mathContainer hidden">class="mathCode">, and conjectured that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16302230&_mathId=si6.gif&_user=111111111&_pii=S0012365X16302230&_rdoc=1&_issn=0012365X&md5=8ae76d592c14d3110c907f389a8ab28f" title="Click to view the MathML source">n≤(d+1)(d+2)/2class="mathContainer hidden">class="mathCode">. Known bilinear dual hyperovals in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16302230&_mathId=si7.gif&_user=111111111&_pii=S0012365X16302230&_rdoc=1&_issn=0012365X&md5=4ce7d68b3bc049e382e75a5e0846a6d7" title="Click to view the MathML source">V((d+1)(d+2)/2,2)class="mathContainer hidden">class="mathCode"> are the Huybrechts dual hyperoval and the Buratti–Del Fra dual hyperoval. In this paper, we investigate on the covering map class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16302230&_mathId=si8.gif&_user=111111111&_pii=S0012365X16302230&_rdoc=1&_issn=0012365X&md5=4015b4508e4663bfbd68105409b7864a">class="imgLazyJSB inlineImage" height="21" width="228" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0012365X16302230-si8.gif">class="mathContainer hidden">class="mathCode">, where the dual hyperovals class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16302230&_mathId=si9.gif&_user=111111111&_pii=S0012365X16302230&_rdoc=1&_issn=0012365X&md5=0917a484dabe48e41d51234d4cb58d26">class="imgLazyJSB inlineImage" height="21" width="92" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0012365X16302230-si9.gif">class="mathContainer hidden">class="mathCode"> and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16302230&_mathId=si10.gif&_user=111111111&_pii=S0012365X16302230&_rdoc=1&_issn=0012365X&md5=67f8c3dcb50cc3daa6b8dc1fdb3a3a7c" title="Click to view the MathML source">Sc(l,GF(2r))class="mathContainer hidden">class="mathCode"> are constructed in Taniguchi (2014). Using the result, we show that the Buratti–Del Fra dual hyperoval has a bilinear quotient in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16302230&_mathId=si11.gif&_user=111111111&_pii=S0012365X16302230&_rdoc=1&_issn=0012365X&md5=f9ed7f106d4206deb81a4274af57f214" title="Click to view the MathML source">V(2d+1,2)class="mathContainer hidden">class="mathCode"> if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16302230&_mathId=si1.gif&_user=111111111&_pii=S0012365X16302230&_rdoc=1&_issn=0012365X&md5=687fe321ae4845dff9bb032121e41e88" title="Click to view the MathML source">dclass="mathContainer hidden">class="mathCode"> is odd. On the other hand, we show that the Huybrechts dual hyperoval has no bilinear quotient in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16302230&_mathId=si11.gif&_user=111111111&_pii=S0012365X16302230&_rdoc=1&_issn=0012365X&md5=f9ed7f106d4206deb81a4274af57f214" title="Click to view the MathML source">V(2d+1,2)class="mathContainer hidden">class="mathCode">. We also determine the automorphism group of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16302230&_mathId=si10.gif&_user=111111111&_pii=S0012365X16302230&_rdoc=1&_issn=0012365X&md5=67f8c3dcb50cc3daa6b8dc1fdb3a3a7c" title="Click to view the MathML source">Sc(l,GF(2r))class="mathContainer hidden">class="mathCode">, and show that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16302230&_mathId=si15.gif&_user=111111111&_pii=S0012365X16302230&_rdoc=1&_issn=0012365X&md5=6f7f4805193c4fda037b5db965767da9" title="Click to view the MathML source">Aut(Sc(l2,GF(2rl1)))<Aut(Sc(l,GF(2r)))class="mathContainer hidden">class="mathCode">.