I'm working on a proof, and I would like to show that if every subgroup is characteristic in G, then any subgroup of a subgroup is also characteristic in that subgroup.
In other words, $H char G, K char G$ and $K \leq H \implies K char H$.
Now I am thinking that this isn't true but I'm not sure why.
Also, I was wondering if we have a group $G$ and a normal subgroup $N$, then if $H char G \implies \overline{H} char \overline{G}$.
Thank you in advance!
