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No. Take a f.g. non-finitely presented group $G$ with trivial $Out(G)$ and trivial center (such groups clearly exist). Suppose that $G$ is a normal subgroup of $H$. Then every $h\in H$ acts on $G$ by conjugation $\phi_g$. Since $Aut(G)=Inn(G)$, there exists $g\in G$ such that $x^g=x^h$ for every $x\in G$. Hence $g^{-1}h$ centralizes $G$. Therefore $H=G Z_H(G)$ where $Z_H(G)$ intersects $G$ trivially and is a normal subgroup. Hence $H$ is a direct product of $G$ and $Z=Z_H(G)$. Since $H$ is finitely generated, $Z$ is finitely generated. Hence if $H$ is finitely presented, then $G$ is presented by the presentation of $H$ plus finitely many relations killing the generators of $Z$, so $G$ must be finitely presented, a contradiction.