  A search is presented for the decay of the 125 GeV Higgs boson (H) to a pair
of new light pseudoscalar bosons (a), followed by the prompt decay of each a
boson to a bottom quark-antiquark pair, H $\to$ aa $\to$
$\mathrm{b\bar{b}b\bar{b}}$. The analysis is performed using a data sample of
proton-proton collisions collected with the CMS detector at a center-of-mass
energy of 13 TeV, corresponding to an integrated luminosity of 138 fb$^{-1}$.
To reduce the background from standard model processes, the search requires the
Higgs boson to be produced in association with a leptonically decaying W or Z
boson. The analysis probes the production of new light bosons in a 15 $\lt$
$m_\mathrm{a}$ $\lt$ 60 GeV mass range. Assuming the standard model predictions
for the Higgs boson production cross sections for pp $\to$ WH and ZH, model
independent upper limits at 95% confidence level are derived for the branching
fraction $\mathcal{B}$(H $\to$ aa $\to$ $\mathrm{b\bar{b}b\bar{b}}$). The
combined WH and ZH observed upper limit on the branching fraction ranges from
1.10 for $m_\mathrm{a} =$ 20 GeV to 0.36 for $m_\mathrm{a} =$ 60 GeV,
complementing other measurements in the $\mu\mu\tau\tau$, $\tau\tau\tau\tau$
and bb$\ell\ell$ ($\ell = $ $\mu$, $\tau$) channels.
