  An $m$-cover of the Hermitian surface $H(3,q^2)$ of $PG(3,q^2)$ is a set
$\mathcal{S}$ of lines of $H(3,q^2)$ such that every point of $H(3,q^2)$ lies
on exactly $m$ lines of $\mathcal{S}$, and $0<m<q+1$. Segre (1965) proved that
if $q$ is odd, then $m=(q+1)/2$, and called such a set $\mathcal{S}$ of lines a
hemisystem. Penttila and Williford (2011) introduced the notion of a relative
hemisystem: a set of lines $\mathcal{R}$ of $H(3,q^2)$, $q$ even, disjoint from
a symplectic subgeometry $W(3,q)$ such that every point of $H(3,q^2)\setminus
W(3,q)$ lies on exactly $q/2$ elements of $\mathcal{R}$. In this paper, we
provide an analogue of Segre's result by introducing relative $m$-covers of
$H(3,q^2)$ with respect to a symplectic subgeometry and proving that $m$ must
necessarily be $q/2$.
