Group homomorphisms inducing $\operatorname {mod}\text {-}p$ cohomology monomorphisms

Let $f\colon G\to K$ be a homomorphism of $p$-groups such that
$f^{(n)}\colon H^n(K,\mathbf Z /p)\to H^n(G,\mathbf Z/p)$ is injective, for $1\le n\le 2$. We prove that the non-bijectivity of $f$ implies the existence of a quotient $L$ of $G$ containing $K$ as a proper direct factor. This gives a refined proof of a result of Evens, which asserts that $f$ is bijective if $f^{(1)}$ is.