The "connecting homomorphism" for the LES of iterated classifying spaces for an SES of topological Abelian groups

Question

• Assume we are working with a "nice" category $\mathrm{Top}$ of topological spaces closed under the categorical constructions we'll use.

• Let $B : \mathrm{TopGrp} \rightarrow \mathrm{Top}_*$ be the (pointed) classifying-space functor.

  We know that for topological Abelian groups $A$ we can place on $BA$ a top Ab grp structure, such that we get an endofunctor $\bar{B}$ on $\mathrm{TopAbGrp}$ which "commutes" with the forgetful functors $\mathrm{TopAbGrp}\hookrightarrow \mathrm{TopGrp} \rightarrow\mathrm{Top}_*$.

• Given an s.e.s. $$ 0\rightarrow K \rightarrow L \rightarrow M \rightarrow 0 $$ in $\mathrm{TopGrp}$, we get a "homotopy-exact" sequence $$ BK \rightarrow BL \rightarrow BM $$ in $\mathrm{Top}_*$.

• Furthermore, by treating $K\rightarrow L \rightarrow M$ as a principal $K$-bundle, we get a continuous map $M \xrightarrow{f} BK$, unique up to "coherent homotopy".

• If the $K,L,M$ are all Abelian groups, then we can refine the second line to $$ \bar{B} K \rightarrow \bar{B} L \rightarrow \bar{B} M $$ as a sequence in the category $\mathrm{TopAbGrp}$, still "homotopy-exact" in $\mathrm{Top}$.

My first question is:

1. Can we also assume the sequence of $\bar{B}$ of $K,L,M$ is exact as a sequence of Abelian groups?

• And, we still have a continuous map $M \xrightarrow{f} BK$ to the underlying topological space of $\bar{B}K$, still unique "up to coherent homotopy".

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My second question is:

2. In the Abelian case, how do we show it is possible to choose the map $M \xrightarrow{f} BK$ such that it is a (topological) Abelian group homomorphism $M\xrightarrow{\phi} \bar{B}K$? (Or, is my assumption incorrect and this is not always possible?)

• If I'm not mistaken, this is used e.g. to get the Bockstein homomorphism, which is the map induced on generalized cohomology by $\bar{B}^nM \xrightarrow{\bar{B}^n \phi} \bar{B}^{n+1} K$ ;

• which is the connecting homomorphism for an LES (exact in the homotopy sense? or in the group sense?) of topological Abelian groups $$ 0 \rightarrow K \rightarrow L \rightarrow M \xrightarrow{\phi} \bar{B} K \rightarrow \bar{B}L \rightarrow \bar{B}M \xrightarrow{\bar{B} \phi} \bar{B}^2 K \rightarrow \cdots $$ where the maps other than the connecting homomorphism are all iterated $\bar{B}$ images of $K \rightarrow L \rightarrow M$.