I am interested in the second (and higher as well) homotopy groups of compact CW complexes. I know these groups don't need to be finitely generated (e.g. for $S^1 \vee S^2$ they are not), but I'd like to know more about the structure of these groups. Below $X$ always denotes a compact CW complex.

For example, it seems unlikely to me that one could have something like $\pi_2(X) \cong \mathbb Q$ or $\pi_2(X) \cong \mathbb Q / \mathbb Z$, but then again I really don't know much about the topic. Here is are some specific questions:

(1) Does $\pi_2$ decompose into torsion-free and torsion parts as follows: $\pi_2(X) \cong \bigoplus_I \mathbb Z \oplus \bigoplus_i \bigoplus_{J_i} \mathbb Z/p_i^{r_i}$, where the $I$ and $J_i$ are index sets, $p_i$ are some primes? In all the examples I have ever seen, this happens. Of course we can't use the structure theorem for finitely generated Abelian groups to conclude something like this, but maybe it is still true?

(2) If the above is false, is the following weaker thing true? If $\pi_2(X) \otimes k = 0$ for every field $k$, then is $\pi_2(X) = 0$?

(3) Do only finitely many different primes occur as the orders elements in $\pi_2(X)$?

I'm aware that via Hurewicz we have $\pi_2(X) \cong H_2(\widetilde X)$, I don't know if this helps.