I need this for a counterexample: the multiplication in the fundamental group $\pi_1(\Sigma X_+)$, when it is equipped with the topology inherited from $\Omega \Sigma X_+$, fails to be continuous for the sort of space in the question, by a result from

J. Brazas, The topological fundamental group and hoop earring spaces, 2009, arXiv:0910.3685

but the author doesn't supply an explicit example of such a space.