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Definition: A subgroup of a group is termed strongly potentially characteristic if there is an embedding of the bigger group in some group such that, in that embedding both the group and the subgroup become characteristic.

One can refer to this link.

Although, I have somewhat understood the definition, I would like to have some examples of such subgroups. Moreover, I would also like to know, as to what's the essence of defining such subgroups in such a manner. Where are they applicable?

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    @Arturo Magidin: Sorry Arturo! I can't think of anything either. I just saw the website and thought MATH.SE would be an ideal place for knowing more about such subgroups.2010-08-28

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Obviously, a characteristic subgroup $H \subseteq G$ is strongly potentially characteristic.

Here is an example of a non-characteristic subgroup. Consider the groups: $H = \{ 0 \} \times \mathbb{Z}/2 \subset \mathbb{Z}/2 \times \mathbb{Z}/2 = G$ The subgroup $H$ is not characteristic in $G$, because the automorphism of $G$ switching the coordinates does not fix $H$.

But now consider the embedding: $ G \to \mathbb{Z}/2 \times \mathbb{Z}/4 = G'$ $ (a,b) \mapsto (a,2b) $ Moreover, consider the characteristic morphism: $ T \colon G' \to G' \quad g' \mapsto 2g'$

Now $G$ is the kernel of $T$, hence it is characteristic. Moreover, $H$ is the image of $T$, hence is characteristic as well.