Can we have this situation? If so, what examples are there of it?

Question

Let $G$ be a group, let $F$ be a normal subgroup of $G$, and let $E$ be a normal subgroup of $F$. That is, let $E$ be a normal subgroup of $F$, and let $F$ in turn be a normal subgroup of $G$. Is $E$ normal in $G$? If not necessarily, then what example illustrates this situation?

Answer 1

Normality is not transitive. Let $A_4$ be the alternating group of 4th degree. Then the subgroup of double transpositions $V = \{ e, (12)(34),(13)(24),(14)(23)\}$ is a normal subgroup and if $Z = \{e,(12)(34)\}$ then $Z$ is a normal subgroup of $V$ that is not normal in $A_4$

Answer 2

Counterexample:

$$\{(1),(12)(34)\}\lhd \{(1),(12)(34),(13)(24),(14)(23)\}\lhd A_4\;,\;\;\text{but}\;\;\{(1),(12)(34)\}\rlap{\;\,/}\lhd A_4$$