A necessary condition is that $H $ be closed under any function $f $ that is equivariant under automorphisms $\sigma $ of $G $: $f(\sigma (x))=\sigma (f (x)) $ (and similarly for functions of several variables).

For instance, taking inverses, products, and $n $th roots. For the latter see also http://groupprops.subwiki.org/wiki/Fixed-point_subgroup_of_a_subgroup_of_the_automorphism_group

A necessary condition is that $H $ be closed under any function $f $ that is equivariant under automorphisms $\sigma $ of $G $: $f(\sigma (x))=\sigma (f (x)) $ (and similarly for functions of several variables).

For instance, taking inverses, products, and $n $th roots. For the latter see also http://groupprops.subwiki.org/w/index.php?title=Fixed-point_subgroup_of_a_subgroup_of_the_automorphism_group#Weaker_properties