Group Cohomology and Change of Coefficients

Question

Let $G$ be a group and $V$ be a representation of $G$ over a field $K$. Let $L$ be any field containing $K$. Consider the natural morphism $$\varphi:H^i(G,V)\otimes_KL\rightarrow H^i(G,V\otimes_KL).$$ My computations seem to show that $\varphi$ is injective, and I was thus wondering if $\varphi$ is also surjective. If not, is there any explicit description on the image or the cokernel of $\varphi$? Thank you so much!