Fully transitive $p$-groups with finite first Ulm subgroup

An abelian $p$-group $G$ is called (fully) transitive if for all $x,y\in G$ with $U_G(x)=U_G(y)$ ( $U_G(x)\leq U_G(y)$) there exists an automorphism (endomorphism) of $G$ which maps $x$ onto $y$. It is a long-standing problem of A. L. S. Corner whether there exist non-transitive but fully transitive $p$-groups with finite first Ulm subgroup. In this paper we restrict ourselves to $p$-groups of type $A$, this is to say $p$-groups satisfying $\mathrm{Aut}(G)\upharpoonright_{ p^{\omega}G} = U(\mathrm{End}(G) \upharpoonright_{p^{\omega}G})$. We show that the answer to Corner's question is no if $p^{\omega}G$ is finite and $G$ is of type $A$.