Subgroup-defining function value is characteristic

Definition
Suppose $$f$$ is a subgroup-defining function, i.e., $$f$$ associates to every group $$G$$ a subgroup $$f(G)$$, with the property that given an isomorphism $$\sigma:G_1 \to G_2$$, the image of $$f(G_1)$$ under $$\sigma$$ is $$f(G_2)$$.

Then, for every group $$G$$, $$f(G)$$ is a proves property satisfaction of::characteristic subgroup of $$G$$.

Conclusion about normality
Combined with the fact that characteristic implies normal, this tells us that any subgroup-defining function value is normal.

Proof
Given: A group $$G$$, a subgroup-defining function $$f$$, an automorphism $$\sigma$$ of $$G$$.

To prove: $$\sigma(f(G)) = f(G)$$.

Proof: Since $$\sigma:G \to G$$ is an automorphism, it is in particular an isomorphism. Thus, the definition of subgroup-defining function above, setting $$G_1 = G_2 = G$$, gives us what we need to prove.