Let $p$ be a prime number. I am interested in the abelian groups $G$ with the following property:

(U) every finite abelian $p$-group $A$ admits a monomorphism $A\hookrightarrow G$.

In other words, every finite abelian $p$-group is a subgroup of $G$.

For instance, denoting by $C_n$ the cyclic group of order $n$, the group $$ G_p:= \bigoplus_{k=1}^\infty C_{p^k} $$ has this property.

I am interested whether (U) is equivalent to the following property:

(U') there is a monomorphism $G_p\hookrightarrow G$.

Clearly, (U')$\Rightarrow$(U), but the opposite implication does not look obvious to me (perhaps it is wrong).

Another related question: does it exist a "nice" characterization of the abelian groups $G$ such that both monomorphisms $G\hookrightarrow G_p$ and $G_p\hookrightarrow G$ exist? Clearly, groups of the shape $$ \bigoplus_{k=1}^\infty C_{p^k}^{m_k}, $$ (where $(m_k)$ is a sequence of non-negative integers with infinitely many strictly positive terms) have this property. Are there others?