Closures of permutation groups with restricted nonabelian composition factors

Given a permutation group $G$ on a finite set $\mathrm{\Omega }$, let $G^{(k)}$ denote the $k$-closure of $G$, that is, the largest permutation group on $\mathrm{\Omega }$ having the same orbits in the induced action on ${\mathrm{\Omega }}^k$ as $G$. Recall that a group is $\mathrm{A}\mathrm{l}\mathrm{t}(d)$-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 $\mathrm{A}\mathrm{l}\mathrm{t}(d)$-free group is again $\mathrm{A}\mathrm{l}\mathrm{t}(d)$-free for $k\ge 4$ and $d\ge 25$.