Are characteristic subgroups "unique" within the containing group?

Question

A normal subgroup is invariant through all inner automorphisms of the original group.

A characteristic subgroup is invariant through all automorphisms, period (inner or outer) of the original group.

An example of a normal but non-characteristic subgroup can be found in the multiplicative Quaternion group $Q$: The subgroup $\{1, -1, i, -i\}$ is normal, but not characteristic. By defining a "re-labelling" automorphism of $Q$ that swaps $i$ with $j$ (and $-i$ with $-j$, of course) we see that is obviously isomorphic to $\{1, -1, j, -j\}$, even though conjugation by no element of the group can achieve this mapping (i.e., it is an outer automorphism).

Is this always the case? In other words, which one of the following is true?

A) If a group $G$ contains a characteristic subgroup $H$, it can not contain any other subgroup different but isomorphic to $H$ ("only one copy").

B) A group $G$ might contain several different subgroups $H, H', H''...$, all isomorphic to each other, but still lack any automorphisms (inner or outer) that map the H between themselves.

It seems to me that the "re-labelling" used to generate the necessary outer automorphism in the Quaternion example (and thus prove the group non-characteristic) should always be achievable, but I am not confident of this.

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EDIT: Thanks everyone for the feedback. Now I see how short-sighted my question is: A group $G$ can potentially have many subgroups isomorphic to each other (e.g., several "copies" of $C_2$), perhaps some of them characteristic, without said isomorphisms necessarily extending to an automorphism of $G$.

Answer 1

Consider the dihedral group of order 8. Its center has order 2 and is characteristic, but it also contains other subgroups of order 2. There are plenty of other examples.

Answer 2

A group can have 2 (or more) isomorphic subgroups, which are both characteristic and what is required is basically that you have 2 (or more) isomorphic subgroups that 'sit uniquely' in the lattice of subgroups. To give an explicit example. Have a look at this subgroup lattice for the semi-direct product $C_4 \rtimes C_4$. All of the ordered pairs are GAP ids for the groups in question. The order 2 groups are identified as $(2,1)$. Notice how they each sit uniquely in the overall lattice (each is a member of a different number of order 4 subgroups!)