Let $G$ be a finite abelian $p$-group, $p$ a prime. I say that a pair $(G',\varphi)$ is a *maximal cyclic quotient* (please excuse me if this definition already exists and refers to a different concept) of $G$ if $G'$ is a cyclic group and $\varphi\colon G\to G'$ is a surjective map with the following property: if $H\leq \ker\varphi$ is a subgroup such that $G/H$ is cyclic, then $H=\ker\varphi$.

My first question is the following: is it true that if $(G',\varphi)$ is a maximal cyclic quotient of $G$ then $G\simeq G'\times\ker\varphi$?

Let now $G=\prod_iG_i$ be the decomposition of $G$ into cyclic $p$-groups and let $H$ be a subgroup of $G$. Does there exist an automorphism of $G$ which maps $H$ to a product $\prod_iH_i$ with $H_i\leq G_i$ for all $i$? An affirmative answer to this question would imply an affirmative answer to the first one, but I have no clue about its truth; it sounds as a pretty strong statement indeed...