A problem in May's Algebraic Topology book (on page 33) is
Suppose $p:H \to G$ is a covering map of topological groups, and let $K < H$ be its kernel. Show that $k \mapsto (g \mapsto kg)$ exhibits an isomorphism between $K$ and the covering automorphisms of $H$.
(As for the solution, it's pretty easy to see directly that these are covering automorphisms, and then easy properties of covering spaces imply that they are the only ones.)
My question is: the solution implies that $p_*(\pi_1(H, e_H)) = p_*(\pi_1(H, k))$ for all $k \in K$. Is this easy enough to see directly, or would we always appeal to an exercise like the above to show this?