  The first fundamental theorem of invariant theory for the orthosymplectic
supergroup ${\rm OSp}(V)$ (where $V$ has superdimension $(m|2n)$) in the
endomorphism algebra setting states that there is a surjective algebra
homomorphism $F_r^r: B_r(m-2n)\rightarrow {\rm{End}}_{{\rm OSp}(V)}(V^{\otimes
r})$ from the Brauer algebra of degree $r$ to the endomorphism algebra of
$V^{\otimes r}$ over ${\rm OSp}(V)$. The second fundamental theorem in this
setting seeks to describe ${\rm Ker} F_r^r$ as a $2$-sided ideal of
$B_r(m-2n)$. We show that ${\rm Ker} F_r^r\neq 0$ if and only if $r\geq
r_c:=(m+1)(n+1)$, and present a basis and a dimension formulae for ${\rm Ker}
F_r^r$. As a 2-sided ideal, ${\rm Ker} F_r^r$ for any $r\ge r_c$ is generated
by ${\rm Ker} F_{r_c}^{r_c}$, for which a set of generators is explicitly
constructed in terms of Brauer diagrams. As applications of these results, we
obtain the necessary and sufficient conditions for the endomorphism algebra
$\rm{End}_{{\mathfrak {osp}}(V)}(V^{\otimes r})$ over the orthosymplectic Lie
superalgebra ${\mathfrak {osp}}(V)$ to be isomorphic to $B_r(m-2n)$, and give
new proofs for the main theorems in recent papers of G. Lehrer and R. Zhang on
the second fundamental theorem of invariant theory for the orthogonal and
symplectic groups.
