  In this article, we study the exponents of metastable homotopy of mod $2$
Moore spaces. Our result gives that the double loop space of $4n$-dimensional
mod $2$ Moore spaces has a multiplicative exponent $4$ below the range of $4$
times the connectivity. As a consequence, the homotopy groups of
$4n$-dimensional mod $2$ Moore spaces have an exponent $4$ below the range of
$4$ times the connectivity.
