The Eilenberg-MacLane space $K(G,n)$ is defined such that $\pi_n(K(G,n)) = G$ and $\pi_i (K(G,n)) = 0$ if $i \neq n$. It is well known that such space is unique up to the homotopy type.
Now we define a space $K=K(G,H)$ for any group $G$ and an abelian group $H$ such that $\pi_1(K(G,H)) = G$, $\pi_2(K(G,H)) = H$ and $\pi_i (K(G,H)) = 0$ if $i \neq 1,2$. My question is, is this space $K(G,H)$ well defined up to homotopy type?
This question was answered in the paper
Mac Lane, S. and Whitehead, J. H.~C. On the 3-type of a complex. Proc. Nat. Acad. Sci. U.S.A. 36 (1950) 41--48,
although nowadays we use the word "2-type". As Mike Miller writes, the classification is in terms of crossed modules. This structure consists of a morphism of groups, say $\mu: M \to P$, together with an action of $P$ on $M$, say $(m,p) \mapsto m^p$, satisfying rules:
CM$1$) $\mu(m^p) = p^{-1}\mu(m) p$;
CM$2$) $n^{-1}mn= m^{\mu n}$;
for all $m,n \in M, p \in P$.
These rules are satisfied by the boundary map $\delta: \pi_2(X,A,x) \to \pi_1(A,x)$ for the relative homotopy group. There is a classifying space functor $\mathbb B$ which assigns to such a crossed module a pair of pointed spaces $(B(M \to P), BP)$ where $BP$ is the classifying space of the group $P$, i.e. is a $K(P,1)$, and the homotopy crossed module assigned to this pair is naturally isomorphic to $\mu: M \to P$. This crossed module determines an element $k \in H^3(Cok \; \mu,Ker \; \mu)$ and these elements determine the pointed homotopy types of the classifying space pairs. Full details of how to do all this, and calculations of and with crossed modules, are in the book partially titled Nonabelian Algebraic Topology (EMS Tracts in Math, Vol 15, 2011), which contains an extensive bibliography. There is a classification of "Homotopy 2-types of low order" by G. Ellis and L.van Luyen in J. Experimental Math. 2014.