When is the diagonal inclusion a $\Sigma_2$-cofibration?

Question

Recall that a space $X$ is called locally equiconnected or LEC if the diagonal map $d:X\hookrightarrow X\times X$ is a cofibration. For example, CW-complexes are LEC. There is some discussion of this concept at this MO question.

Let $G$ be a finite group. Recall that a $G$-map $i: A\to Y$ is called a $G$-cofibration if it has the $G$-homotopy extension property with respect to all $G$-maps $f: Y\to Z$.

If $X$ is LEC, is the diagonal map $d: X\hookrightarrow X\times X$ a $\Sigma_2$-cofibration, where the symmetric group acts trivially on $X$ and by permuting factors on $X\times X$?

I've tried searching the literature on equivariant homotopy theory, but this doesn't seem to fall quite in that territory since $X$ itself does not come equipped with a group action.

Answer 1

I ended up needing this again, and eventually proved that $d:X\hookrightarrow X\times X$ is a $\Sigma_2$-cofibration as long as $X$ is an ENR. Although this doesn't quite answer the original question as asked, it seems general enough to be useful, so I thought I'd post it here.

More generally, if $G$ is a finite group, $X$ is a $G$-ENR, and $A\subseteq X$ is a closed sub-$G$-ENR, then the inclusion $A\hookrightarrow X$ is a $G$-cofibration.

The reference: Proposition 2.7 and Corollary 2.8 of https://arxiv.org/abs/1703.07142.