A group may have two distinct characteristic Sylow-$p$-subgroups

Question

There is a statement:

A group may have two distinct characteristic Sylow-$p$-subgroups.

I think it is not necessarily true because if there is unique Sylow-$p$-subgroups, it is characteristic; if the group is characteristic, it is unique Sylow-$p$-subgroup.

Can anyone give some details about this?

Answer 1

The Sylow p-subgroups of $G$ are all conjugate. So if there are > 1 such subgroups, conjugation by some element of $G$ sends one Sylow p-subgroup to another. Conjugation by a fixed element of $G$ is an automorphism.

Answer 2

The question could also be interpreted differently: it is not clear if we are talking about the same prime $p$ - of a nilpotent group with $|G|$ having more than one prime divisor, all its different Sylow subgroups are characteristic.