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Is there a finite group which is contained in infinitely many core-free finite groups as the centralizer of a central involution?

The Brauer–Fowler results show that if a finite group has no non-identity normal subgroups of odd order (so is core-free), then its order is bounded by a function of the size of a few centralizers (of involutions, of strongly real elements, etc.). In particular, the size of a core-free group is bounded by a function of the size of its largest and smallest involution centralizers, and a simple group by a function of the size of any of its involution centralizers.

Sometimes one can apparently do better: there are only finitely many core-free groups with a central involution centralizer of size $4$.

Is there an explicit example where one cannot do better? In other words:

Is the some finite group $X$ such that there are infinitely many finite groups $G$ with no non-identity normal subgroups of odd order, and with a Sylow $2$-subgroup $P$ and involution $t$ in $Z(P)$ such that $C_G(t) = X$?

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Perhaps math overflow would be better for this question? –  Jim Belk Feb 10 '11 at 3:12

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