  We shall first present an explicit realization of the simple $N=4$
superconformal vertex algebra $L_{c} ^{N=4}$ with central charge $c=-9$. This
vertex superalgebra is realized inside of the $ b c \beta \gamma $ system and
contains a subalgebra isomorphic to the simple affine vertex algebra $L_{A_1}
(- \tfrac{3}{2} \Lambda_0)$. Then we construct a functor from the category of
$L_{c} ^{N=4}$--modules with $c=-9$ to the category of modules for the
admissible affine vertex algebra $L_{A_{2} } (-\tfrac{3}{2} \Lambda_0)$. By
using this construction we construct a family of weight and logarithmic modules
for $L_{c} ^{N=4}$ and $L_{A_{2} } (-\tfrac{3}{2} \Lambda_0)$. We also show
that a coset subalgebra of $L_{A_{2} } (-\tfrac{3}{2} \Lambda_0)$ is an
logarithmic extension of the $W(2,3)$--algebra with $c=-10$. We discuss some
generalizations of our construction based on the extension of affine vertex
algebra $L_{A_1} (k \Lambda_0)$ such that $k+2 = 1/p$ and $p$ is a positive
integer.
