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We study some generalized metric properties on the hyperspace $ℱ\left(X\right)$ of finite subsets of a space $X$ endowed with the Vietoris topology. We prove that $X$ has a point-star network consisting of (countable) $wcs$-covers if and only if so does $ℱ\left(X\right)$. Moreover, $X$ has a sequence of $wcs$-covers with property $\left(P\right)$ which is a point-star network if and only if so does $ℱ\left(X\right)$, where $\left(P\right)$ is one of the following properties: point-finite, point-countable, compact-finite, compact-countable, locally finite, locally countable. On the other...