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I am reading Hillman's book "algebraic invariants of links" and on page 20 he mentions the following universal coefficient spectral sequence.

Let X be a connected finite CW complex.Let $H$ be a normal subgroup of $G=\pi_1(X)$ and let $J$ be a normal subgroup of $G$ containing $H$, there is a spectral sequence relating the homology of the covering spaces $X_H$ and $X_J$

$E_2^{pq} = Tor_p (H_q(X;R[G/H]),R[G/J]) \rightarrow H_{p+q}(X;R[G/J]),$

where the local coefficient notation is used: $H_q(X;R[G/H]=H_q(X_H)$.

My questions are the following. Let $f: X \longrightarrow Y$ be a map. Assume $f$ lifts both to maps $f_H : X_H \longrightarrow Y_H $ and $f_J : X_J \longrightarrow Y_J$.
1)Is it correct that, via $f_H$, $f$ induces a map $E_r^{pq}(X) \longrightarrow E_r^{pq}(Y)$ for each $p,q,r$?
2) Assuming the spectral sequence collapses nicely and that the answer to 1) is yes, is there any relation between the map "f_H on the infinity page" and $f_J$?

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    $\begingroup$ All "natural" things are natural. For a cellular map this is even obvious, as you get a map at the level of complexes. $\endgroup$
    – Alex Degtyarev
    Commented Jan 19, 2015 at 21:15
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    $\begingroup$ To add to the comment by @AlexDegtyarev the answer to 2) is that the map "f_H on the infinity page" is the associated graded map of $f_J$. This is always the case when a spectral sequence converges. $\endgroup$
    – user43326
    Commented Jan 20, 2015 at 10:43

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