Extensions for supersingular representations of $G{L}_2(Q_p)$

Let $p>2$ be a prime number. Let $G:=G{L}_2(Q_p)$ and $\pi$, $\tau$ smooth irreducible representations of $G$ on ${\bar{F}}_p$-vector spaces with a central character. We show if $\pi$ is supersingular then $Ex{t}_G^1(\tau ,\pi )\mathrm{\ne }0$ implies $\tau \cong \pi$. This answers affirmatively for $p>2$ a question of Colmez. We also determine $Ex{t}_G^1(\tau ,\pi )$, when $\pi$ is the Steinberg representation. As a consequence of our results combined with those already in the literature one knows $Ex{t}_G^1(\tau ,\pi )$ for all irreducible representations of $G$.