  We present the formalism for connecting a second-order electroweak
$2\xrightarrow[]{H_I+H_I}2$ transition amplitudes in the finite volume (with
two hadrons in the initial and final states) to the physical amplitudes in the
infinite volume. Our study mainly focus on the case where the low-lying
intermediate state consists of two scattering hadrons. As a side product we
also reproduce the finite-volume formula for $2\xrightarrow[]{H_I}2$
transition, originally obtained by Brice\~no and Hansen. With the available
finite-volume formalism, we further discuss how to treat with the finite-volume
problem in the double beta decays $nn\to pp ee\bar{\nu}\bar{\nu}$ and $nn\to pp
ee$.
