This question follows on from a previous one, which has been answered in the negative: Is a locally compact Hausdorff quotient of a locally compact $\sigma$-compact first countable Hausdorff space always first countable?
Let $Y$ be a locally compact, $\sigma$-compact, first countable Hausdorff space and $q:Y\to X$ a quotient map with $X$ Hausdorff. Suppose that $X$ is locally compact. Is $X$ a Frechet-Urysohn space?
I describe an example here including the argument why the quotient space (there called $Y$ is not Fréchet-Urysohn (namely the map $q$ is not hereditarily quotient). The space we take a quotient of is a $\sigma$-compact and locally compact subspace of $\Bbb R$, so fits the bill.
The Arens space $S_2$ (which is Hausdorff), described here as a quotient of a countable disjoint sum of convergent sequences (clearly a $\sigma$-compact locally compact first countable Hausdorff space), and is a classic example of a sequential but not Fréchet-Urysohn space.