Stirling number

Stirling number

I am trying to evaluate the following finite sum:
$$ \sum_{k=1}^{n}(-1)^{k}(k-1)!S(n, k)(\sum_{i=0}^{k}H_{i}), $$ where
$S(n, k)$ are the Stirling's numbers of the second kind and $H_{i}$
denotes the $i$ harmonic number. Could you please shed some light?