We consider the following notion of largeness for subgroups of $S_{\infty }$. A group G is large if it contains a free subgroup on generators. We give a necessary condition for a countable structure A to have a large group Aut(A) of automorphisms. It turns out that any countable free subgroup of $S_{\infty }$ can be extended to a large free subgroup of $S_{\infty }$, and, under Martin’s Axiom, any free subgroup of $S_{\infty }$ of cardinality less than can also be extended to a large free subgroup of $S_{\infty }$. Finally, if Gₙ are countable groups, then...