In Davis & Kirk's Lecture Notes in Algebraic Topology, Exercise 159 reads:
Use the spectral sequence of the universal cover to show that for a path-connected space $X$ the sequence $$\pi_2(X)\stackrel{\rho}{\to} H_2(X)\to H_2(\pi_1(X))\to 0$$ is exact.
Here $\rho$ is the Hurewicz map and the (homology) s.s. of the universal cover has $E^2_{p,q}=H_p(\pi_1(X);H_q(\tilde{X}))\Rightarrow H_{p+q}(X)$.
So far I have a short exact sequence $$0\to \pi_2^*(X)/d_3(H_3(\pi_1(X)))\to H_2(X)\to H_2(\pi_1(X))\to 0$$ By $\pi^*_2(X)$ I mean the quotient of $\pi_2(X)$ by the action of the fundamental group. This is $E^2_{0,2}$ in our sequence by some covering space theory. $H_2(\pi_1(X))$ survives all the way to $E^\infty$ because the whole row $E^2_{p,1}$ is zero since $\tilde{X}$ is connected.
So, my question is this: is the kernel of $\rho:\pi_2^*(X)\to H_2(X)$ simply $d_3(H_3(\pi_1(X)))$, and how can I see this? Thanks for your help!