Under what conditions the End-compactification is metrizable

Suppose that $$X$$ is a hemicompact space, connected and locally connected. In that case, it seems that it is possible to define a "End-compactification" of $$X$$ (in the sense of Freudenthal).

Suppose also that $$X$$ is metrizable. Under what condition on $$X$$, we will have the End-compactification metrizable ? Is it enough if $$X$$ is second-countable ?

In the book Dimension and Extensions by Aarts and Nishiura you will find a theorem giving sufficient conditions that the Freudenthal compactification has the same weight as the starting space. Theorem 3.15 and Corollary 3.16 (pages 276/277): if $$X$$ is rim-compact and the quasi-component space of $$X$$ is compact then $$w(FX)=w(X)$$. In your case, if $$X$$ is metrizable then it is also locally compact and hence rimcompact, and second-countable too; connectivity says that the quasi-component space consists of one point. Hence $$FX$$ is second-countable and therefore metrizable.