Quasi-polynomial time algorithm for permutation group isomorphism
The answer is basically yes, if we take into account the hidden assumption that the degree of the group is at most $n$. More precisely, let $G_1, G_2$ with $|G_1|=| G_2|=n$ be finite permutation groups acting on a set $\Omega$. Then existence of a permutational isomorphism between $G_1$ and $G_2$ can be checked in time $n^{\log_2 n} \cdot {\max(|\Omega|,n)}^{O(1)}$.

As suggested in the question, a permutation group has a base and corresponding strong generating set (SGS) of size at most $\log_2 n$, and both can be computed in time polynomial in $n$. But this isn't even needed, any generating set of size at most $\log_2 n$ is enough.

After computing such a generating set for $G_1$, the algorithm simply maps successively each element of the generating set to an element of $G_2$, and checks for each such mapping whether it leads to an isomorphism of $G_1$ and $G_2$. For those who do, the algorithm then checks whether there is a permutation of $\Omega$ inducing that isomorphism (for example by a canonical labeling technique, which is closely related to a similar technique for deterministic finite automata, but more group theoretic methods also work). Those checks can be done in time ${\max(|\Omega|,n)}^{O(1)}$, and because there are at most $n^{\log_2 n}$ such mappings, and every isomorphism between $G_1$ and $G_2$ is determined by such a mapping, the above time bound follows.