Elements of orders $2k$, for $k\geq 5$ in a semidirect product

Question

Let $G$ be a non-solvable group, $N$ be an abelian 2-subgroup of $G$ such that $N=C_G(N)$ and $G/N\cong Sz(8)$. Does $G$ has elements of orders $2k$, for $k\geq 5$?

Answer 1

There is an irreducible module $M$ of dimension $12$ for $H := {\rm Sz}(8)$ over ${\mathbb F}_2$ in which the elements of order $5$, $7$ an $13$ all act fixed-point-freely. So, if we let $G$ be the semidirect prodcut of $M$ with $H$, then $G$ satisfies your conditions. In fact, the orders of elements of $G$ are $1, 2, 4, 5, 7, 8, 13$.

Alternatively, let $L$ be the direct sum of $M$ and the trivial module for $H$, so ${\rm dim}(L)=13$. Then the semidirect product of $L$ with $H$ has elements of order $10,14$ and $26$.