What class of abelian gropus $\cal K$ is this a very natural one:
the members of $\cal K$ are of the following form:
the direct sum $\oplus$ of finitely many $\mathbb Z_n$' s and finitely many $\mathbb Q$ ?
How this $\cal K$ is called ?
I believe that these are either projective or injective $\mathbb Z-$modules, but I'm not sure.
EDIT: are these $\mathbb Z-$modules actually all $\mathbb Z-$modules ?
A $\mathbb{Z}$-module is the same thing as an abelian group. The class $\mathcal{K}$ that you've described certainly doesn't contain all of the abelian groups (for instance, it doesn't contain $\mathbb{Z}$), so the answer to the question from the edit is "no".
Assuming that your $\mathbb{Z}_n$ is the integers mod $n$, then $\mathbb{Z}_2$ is neither projective nor injective as a $\mathbb{Z}$-module ($A$ be a nonzero finite abelian group then $A$ is not a projective or injective $\Bbb Z$ module.).
I don't know if $\mathcal{K}$ has a standard name - perhaps someone else can address that part of the question.