Let $G$ be a finite group with a central subgroup $Z$ of order 2. Suppose that $Z$ has a direct factor in some (and hence any) 2-Sylow subgroup of $G$. Does this imply that $Z$ has a direct factor in $G$?

I have checked some families of examples I can discard. First, the result is true when $G$ is abelian, or more generally nilpotent, and even more generally when $G$ has a normal 2-Sylow. Second, having in mind that $G$ has even order, the result is very easy when $|G|=2$ (mod 4) and is also true when $|G|=4$ (mod 8). This reduces to the case when 8 divides $|G|$. Third, I also checked it when $|G|=2^np$ with $p$ prime. The first values for $|G|$ that are not covered are thus $72=8\times 9$, $120=8\times15$, $144=16\times9$... A simple argument also shows that a minimal counterexample has no nontrivial normal subgroup of odd order.

I also do not know in the case when $G/Z$ is simple. To discard the first few cases, for $G=\mathrm{SL}_2(\mathbf{F}_q)$, $q$ odd prime power, which has center of order 2, the center is not a direct factor in the 2-Sylow since $-I_2$ is square of an element of order 4.