Obstruction theory for non-simple spaces

Paul Olum developed some obstruction theory for maps into non-simple spaces back in the 1940-ies and 50-ies. You may want to check out his paper "Obstructions to extensions and homotopies", Annals of Mathematics, Vol 52, 1950, pp 1-50, if you have not looked at it yet.

@Evans Jenkins: A comparison of the work of Whitehead and Olum is given in

Ellis, G.J. "Homotopy classification the J.H.C. Whitehead way". Exposition. Math. 6 (1988) 97--110.

He writes (and I leave the reader to find the citations):

``In view of the ease with which Whitehead's methods handle the classifications of Olum and Jajodia, it is surprising that the papers \cite{Olum53} and \cite{Jaj80} (both of which were written after the publication of \cite{W49:CHII}) make respectively no use, and so little use, of \cite{W49:CHII}.

``We note here that B. Schellenberg, who was a student of Olum, has rediscovered in \cite{Sch73} the main classification theorems of \cite{W49:CHII}. The paper \cite{Sch73} relies heavily on earlier work of Olum.''

Whitehead used what he calls "homotopy systems", which we now call "free crossed complexes"; the notion of crossed complex goes back to Blakers in 1948 (Annals of Math), and a full account is in the 2011 EMS Tract Vol 15 Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy grouopids. The relation between crossed complexes and chain complexes with operators is quite subtle; it was first developed by Whitehead, and in CHII he explains, in our terms, that crossed complexes have better realisation properties that chain complexes with operators. For example, the latter do not model homotopy 2-types.

Section 12.3 of the above Tract is on the homotopy classification of maps, including the non simply connected case, and Section 12.4 is on local systems, but it may be that your example is out of reach of the "linear" theory of crossed complexes. The homotopy classification of $3$-types requires quadratic information, see books by Baues and also

Ellis, G.J. "Crossed squares and combinatorial homotopy". Math. Z. (214} (1993) 93--110.

So there is sill a lot of work to be done!