  We unravel the ground state properties and the non-equilibrium quantum
dynamics of two bosonic impurities immersed in an one-dimensional fermionic
environment by applying a quench of the impurity-medium interaction strength.
In the ground state, the impurities and the Fermi sea are phase-separated for
strong impurity-medium repulsions while they experience a localization tendency
around the trap center for large attractions. We demonstrate the presence of
attractive induced interactions mediated by the host for impurity-medium
couplings of either sign and analyze the competition between induced and direct
interactions. Following a quench to repulsive interactions triggers a breathing
motion in both components, with an interaction dependent frequency and
amplitude for the impurities, and a dynamical phase-separation between the
impurities and their surrounding for strong repulsions. For attractive
post-quench couplings a beating pattern owing its existence to the dominant
role of induced interactions takes place with both components showing a
localization trend around the trap center. In both quench scenarios, attractive
induced correlations are manifested between non-interacting impurities and are
found to dominate the direct ones only for quenches to attractive couplings.
