  The following result has been shown recently in the form of a dichotomy: For
every total clone $C$ on $\mathbf{2} := \{0,1\}$, the set $\mathcal{I}(C)$ of
all partial clones on $\mathbf{2}$ whose total component is $C$, is either
finite or of continuum cardinality. In this paper we show that the dichotomy
holds, even if only strong partial clones are considered, i.e., partial clones
which are closed under taking subfunctions: For every total clone $C$ on
$\mathbf{2}$, the set $\mathcal{I}_{\mathrm{Str}}(C)$ of all strong partial
clones on $\mathbf{2}$ whose total component is $C$, is either finite or of
continuum cardinality.
