Continuous action on tensor product

Question

Let $G$ be a profinite group and $V,W$ be $k$-vector spaces with discrete topology. Suppose $G$ acts continuously on $V$ and $W$, we extend the action of $G$ to $V \otimes_k W$ by defining on simple tensors as $g.(v \otimes w)=g.v\otimes g.w$. Does $G$ act continuously on $V\otimes_k W$ which is endowed with discrete topology.