Take the tour ×  In a forthcoming paper with Venkatesh and
Westerland, we require the following funny definition. Let G be
a finite group and c a conjugacy class in G. We say the pair
(G,c) is nonsplitting if, for every subgroup H of G, the
intersection of c with H is either a conjugacy class of H or is
empty.  For example, G can be the dihedral group of order 2p and
c the class of an involution.  The case where c is an involution
is o special interest to us. One way to construct nonsplitting
pairs is by taking G to be a semidirect product of N by (Z/2^k
Z), where N has odd order, and c is the conjugacy class
containing the involutions of G. Are these the only examples? In
other words:  Question 1: Is there a nonsplitting pair (G,c)
with c an involution but where the 2-Sylow subgroup of G is not
cyclic?  Slightly less well-posed questions:  Question 2: Are
there "interesting" examples of nonsplitting pairs with c not an
involution? (The only example we have in mind is G = A_4, with c
one of the classes of 3-c