Let $P_n$ denote the pro-$p$ completion of $F_n$ the free group of rank $n$. Given a (abstract) group homomorphism

$$
\phi:P_n\rightarrow G
$$

where $G$ is a discrete group. Is $\phi$ continuous?

The case $G$ is finite is a Theorem of Serre and $\phi$ is continuous in this case.