Thank you, this was quick and helpful!! Could you please add some reference or say a couple more words about "have structural significance; for instance they arise as corrections to the canonical heights of sections of an elliptic surface." Obviousely I'll quote you@MO ;-)

Does it have to do with the subgroup in $\Lambda^*/\Lambda$ generated by $[p]$?

Thank you very much, that was the right paper! And with little effort (above) I could show both questions are equivalent, hence you earned definitely your bounty ;-) BUT I would aprechiate another reference, since googling "distinguished cocycle" was not very successful.

Ah, got it, thanx! (I thought it was in the non-prime case just not sufficient...anyway the quick check was instructive)

Thank you, that was already very helpful. I started to check explicitly in the Atlas, that the statement still holds for the exceptions in the list $\mathbb{A}_6,\mathbb{A}_7,A_3(4),{^2}A_3(3^2),{^2}E_6(2^2),M_{22},Fi_{22}$, which is not so hard given the explicit projective character table given there....However still no counterexamples

I understand you wonder about the 2-cocycle without extension interpretation: In case $C_2\times C_2=\langle g,h\rangle$ there is indeed a unique nontrivial cohomology class. Say the 2-cocycle $\sigma(g,g)=\sigma(h,h)=1, \sigma(g,h)=1,\sigma(h,g)=-1$; any other $\sigma$ in this class will have $\sigma(g,h)\sigma(h,g)=-1$, so all are nonsymmetric.