A slight expansion on my comment, sort of complimentary to Tom's response.
In complete generatlity $H_2X$ tells you nothing about $\pi_1 X$.
If $X = A \times B$ with $A$ a $K(\pi,1)$ and $B$ a $K(\pi,2)$, provided $H_2(A)=0$, you have that $H_2 X = H_2 B$.
Since there are lots of $K(\pi,1)$ spaces with $H_2$ trivial, this allows you to construct many spaces with identical fundamental groups yet $H_2$ varies wildly.
You'll want to restrict to fairly particular spaces to avoid this independence.