$H^2(A,\mathbb{Z}_{(p)})$

Question

This is not familiar ground to me, so I apologise in advance for possible inaccuracies or if the question is trivial. Let $\mathbb{Z}_{(p)}$ be the localisation of the integers at some $(p)$, viewed as an abelian group and $A$ be an abelian group acting trivially on $\mathbb{Z}_{(p)}$. I would be interested to know what the extensions of $A$ by $\mathbb{Z}_{(p)}$ are. Since $\mathbb{Z}_{(p)}$ is almost divisible, having in mind Baer's criterion, I would expect $H^2(A,\mathbb{Z}_{(p)})$ to have a simple description. I am interested in the case where $A$ has no $p$-torsion elements. Is it true then that $H^2(A,\mathbb{Z}_{(p)})=\{id\}$ ? The reason I impose this condition on $A$ is because of the the non-splitting exact sequence: $$0\to \mathbb{Z}_{(p)}\to \mathbb{Q}\to \mu _{p^{\infty}}\to 0$$ If my statement is false, are there some simple conditions on $A$ such that $H^2(A,\mathbb{Z}_{(p)})=\{id\}$ ? Also any relevant source from which I can study related material would be very much appreciated.