Let $\pi_1, \pi_2$ be two irreducible complex representations of $G=\mathrm{GL}_2(\mathbb{Q}_p)$ and assume that there exists a non-split extension $0\to\pi_1\to \pi\to\pi_2\to0$ of representations of $G$. Assume that the central character of $\pi_1$ is $\phi$, then I'd like to show that $\phi$ is also the central character of $\pi_2$.

Idea: For all $z$ in the centre $Z$ of $G$, we can define a function $h_v$ on the underlying space of $\pi$ by $h_v: v\mapsto \phi(z)v-zv$. If $h_v=0$ for all $v$, then the centre of $G$ acts via $\phi$ on $\pi$, in particular the centre of $G$ acts via $\phi$ on $\pi_2$, which is the quotient of $\pi$. If $h_v\neq0$ for some $v_0$, i.e. there exists $z_0$ and $v_0\in\pi$ such that $\phi(z_0)v_0-z_0v_0\neq0$. In this case, I'd like to know how to deduce that $\pi\cong\pi_2$ and in particular, $\pi_2$ has $\phi$ as central character.