A finite group of even order with no normal subgroup of index 2. Let $G$ be a finite group of even order and suppose that there exist an element $x\in Z(G)$ of order $2$, i.e $x$ is in the center of $G$ and satisfying $x^2=1$, $x\neq1$.
Is it possible that $G$ does not contains a normal subgroup of index $2$? I guess the answer is yes but I fail to find any examples of such groups.
Take the binary tetrahedral group $2T\cong Q_8\rtimes C_3$.  It clearly has a central element of order $2$.  And it doesn't have an index-2 subgroup (as it must take the form $C_4\rtimes C_3\subset Q_8\rtimes C_3$ but none of the $C_4$ in $Q_8$ is actually normalised by $C_3$).
The alternating groups, $\frak{A}_n$ do not have an index $2$ subgroup for $n\ge 4$.  The alternating group on $4$ letters is the smallest example, with only $12$ elements.