Let $A$ be a suspension and $X$ be a space with Lusternik-Schnirelmann category $n$ and let $\alpha: A\to X$. It is easy to see that the cofiber $C_\alpha$ has $\mathrm{cat}(C_\alpha) \leq n+1$. One of the standard ways to decide if $\mathrm{cat}(C_\alpha)$ is actually equal to $n+1$ is to look at the Berstein-Hilton-Hopf invariants $H_\sigma(\alpha)$ for the various structure maps $\sigma:X\to G_n(X)$ witnessing the inequality $\mathrm{cat}(X)\leq n$. If $H_\sigma(\alpha) = *$ for any $\sigma$, then $\mathrm{cat}(C_\alpha) \leq n$.
I know that examples are exist to show that the converse implication is false, but my list is very short: in fact, it is limited to the paper `Cogroups which are not suspensions' by Berstein and Harper (LNM 1370).
Question/Request: Other examples of this phenomenon, especially references to examples in print.
