Definition. An isomorphism $\phi: G \to G$ of a group $G$ with itself is an automorphism of G.
Why is it that rephrasing this definition with "An automorphism of a group G is an isomorphism mapping $G$ into $G$" is incorrect while "An automorphism of a group G is an isomorphism mapping $G$ onto $G$" is correct?
Does 'into' pertains to injectivity in the former case and 'onto' means surjectivity in the latter one? But is it not enough to include the term 'isomorphism' to mean that the mapping is both injective and surjective?
I don't think either variant, "mapping $G$ into $G$: or "mapping $G$ onto $G$" is good/correct. Simply "isomorphism of $G$ to $G$". This includes, quite explicitly, both injectivity and surjectivity.
Sometimes people accidentally blur the distinctions between the mathematical and colloquial senses of "to", "onto", "into", etc. It is more reliable to use "injective", "surjective", and "bijective", since these have no colloquial senses. :)