Let $φ:G \rightarrow H$ be a homomorphism. If $K\unlhd H$ prove $\varphi^{-1}[K]=\{x\in G\mid \varphi(x)\in K\}$ is a normal subgroup of $G$.

Question

Let $\varphi:G \rightarrow H$ be a homomorphism. If $K$ is a normal subgroup of $H$, prove: $\varphi^{-1}[K]=\{x\in G\mid \varphi(x)\in K\}$ is a normal subgroup of $G$.

First we need to prove that it is a subgroup of $G$. Obviously $e\in \varphi ^{-1}(K)$

For $x_1, x_2 \in \varphi^{-1}(K)$

$\varphi(x_1)*\varphi(x_2)=\varphi(x_1*x_2) \in K \implies \varphi(x_1*x_2) \in \varphi^{-1}(K) $.

I would appreciate if someone could help me do the last part of the subgroup so to prove that there is a $x^{-1}\in \varphi^{-1}[K]$ for every $x\in \varphi^{-1}[K]$ and obviously the actual proof for the normal subgroup.