at.algebraic topology - Obstruction theory for non-simple spaces

@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, 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!