Closures of permutation groups with restricted nonabelian composition factors

Given a permutation group $G$ on a finite set $\Omega$, let $G^{(k)}$ denote the $k$-closure of $G$, that is, the largest permutation group on $\Omega$ having the same orbits in the induced action on $\Omega^k$ as $G$. Recall that a group is $\mathrm{Alt}(d)$-free if it does not contain a section isomorphic to the alternating group of degree $d$. Motivated by some problems in computational group theory, we prove that the $k$-closure of an $\mathrm{Alt}(d)$-free group is again $\mathrm{Alt}(d)$-free for $k \geq 4$ and $d \geq 25$.