In any context where isomorphisms are defined.
For example, if $G$ and $H$ are two isomorphic groups, then there exists an isomorphism mapping their identity elements together. That is to say, their identity elements are _____, where _____ is the word I'm looking for.
Example two: a graph is vertex transitive if and only if its vertices are pairwise _____.
In general, if $G$, $H$ are groups, there does not exist an isomorphism between them. However, maybe you are looking for the word characteristic.
A characteristic thingie is such that it is unchanged under automorphisms. For example
are characteristic (elements, subgroups, subsets) of $G$. On the other hand, e.g. $p$-Sylow groups are in general not characteristic (because there can be several, i.e. there is not "the" Sylow group).
A graph is vertex transitive if and only if its vertices are pairwise in the orbit of one another under the operation of the automorphism group(!?) In a way, looking at the operation of the automorphism group on (elements, subgroups, ...) of $G$, characteristic (elements, subgroups, ...) are those having one-point orbits.
