What would be a counterexample of $N_G(T)\not\subset N_G(S)$?

Question

Let $G$ be a group and $S,T$ be subgroups of $G$ such that $S\subset T$.

Is there an example such that $N_G(T)\not\subset N_G(S)$?

Also, what is an example such that $N_G(S)\not\subset N_G(T)$?

Answer 1

For the first problem, consider $T=G$ and $S$ is a subgroup of $G$ such that $N(S)\neq G$. I think there are so many examples of this kind.

For the second problem, let $G=S_4$, and $T$ is the subgroup of permutations among $\{1,2,3\}$, which is isomorphic to $S_3$. $S=\{e,(1,2)\}$ is a subgroup of $T$. Then $(3,4)\in N(S)$ but $(3,4)\notin N(T)$ since $$(3,4)(1,2,3)(3,4)=(1,2,4)\notin T.$$