Let
V
V
be a vertex operator superalgebra with the natural order 2 automorphism
σ
\sigma
. Under suitable conditions on
V
V
, the
σ
\sigma
-fixed subspace
V
0
¯
V_{\bar 0}
is a vertex operator algebra and the
V
0
¯
V_{\bar 0}
-module category
C
V
0
¯
\mathcal {C}_{V_{\bar 0}}
is a modular tensor category. In this paper, we prove that
C
V
0
¯
\mathcal {C}_{V_{\bar 0}}
is a fermionic modular tensor category and the Müger centralizer
C
V
0
¯
0
\mathcal {C}_{V_{\bar 0}}^0
of the fermion in
C
V
0
¯
\mathcal {C}_{V_{\bar 0}}
is generated by the irreducible
V
0
¯
V_{\bar 0}
-submodules of the
V
V
-modules. In particular,
C
V
0
¯
0
\mathcal {C}_{V_{\bar 0}}^0
is a super-modular tensor category and
C
V
0
¯
\mathcal {C}_{V_{\bar 0}}
is a minimal modular extension of
C
V
0
¯
0
\mathcal {C}_{V_{\bar 0}}^0
. We provide a construction of a vertex operator superalgebra
V
l
V^l
for each positive integer
l
l
such that
C
(
V
l
)
0
¯
\mathcal {C}_{{(V^l)_{\bar 0}}}
is a minimal modular extension of
C
V
0
¯
0
\mathcal {C}_{V_{\bar 0}}^0
. We prove that these modular tensor categories
C
(
V
l
)
0
¯
\mathcal {C}_{{(V^l)_{\bar 0}}}
are uniquely determined, up to equivalence, by the congruence class of
l
l
modulo 16.