  Let $L$ be a Lie superalgebra over a field of characteristic different from
$2,3$ and write $\mathrm{ID}^{*}(L)$ for the Lie superalgebra consisting of
superderivations mapping $L$ to $L^{2}$ and the central elements to zero. In
this paper we first give an upper bound for the superdimension of
$\mathrm{ID}^{*}(L)$ by means of linear vector space decompositions. Then we
characterize the $\mathrm{ID}^{*}$-superderivation superalgebras for the
nilpotent Lie superalgebras of class 2 and the model filiform Lie superalgebras
by methods of block matrices.
