TITLE: Conjugacy classes in finite groups that remain conjugacy classes when restricted to proper subgroups
QUESTION [23 upvotes]: 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-cycles.)
Question 3:  Does this notion have any connection with anything of pre-existing interest to people who study finite groups?
Update:  Very good answers below already -- I should add that, for maximal "interestingness," the conjugacy class c should generate G.  (This eliminates the examples where c is central in G, except in the case G = Z/2Z).

REPLY [2 votes]: If your $c$ is a conjugacy class of elements of odd prime order $p$, then (using the classification
of finite simple groups), it is still the case that $G = O_{p^{\prime}}(G)C_{G}(x)$ for 
each $x \in c$, where $O_{p^{\prime}}(G)$ denotes the largest normal subgroup of $G$ of order prime to $p$. This might be considered as an odd analogue of Glauberman's $Z^{\ast}$-theorem.
If anyone could come up with a classification-free proof of such a result, it would be of
considerable interest (it is relatvely easy to prove this directly when $G$ is solvable
(or, more generally, $p$-solvable)). Incidentally, your question seems to be related to the
old concept of pronormality (which is treated, for example, in Gorentstein's book
"Finite Groups"). Back to the case $p = 2$, interesting generalizations of Glauberman's
$Z^{\ast}$-theorem were given by D. Goldschmidt and also by E. Shult. One result that Shult
proved which you might find interesting is in a Bull AMS paper (circa 1966), in which he showed
that an element of order $p$ in a finite group which commutes with none of its other conjugates
AND centralizes every $p^{\prime}$-group it normalizes, is central in $G$.