  Core-collapse supernovae are found in galaxies with ongoing star-formation.
In a starburst galaxy hosting an active galactic nucleus with a relativistic
jet, supernovae can take place inside the jet. The collision of the supernova
ejecta with the jet flow is expected to lead to the formation of an interaction
region, in which particles can be accelerated and produce high-energy emission.
We study the non-thermal radiation produced by electrons accelerated as a
result of a supernova explosion inside the jet of an active galactic nucleus
within a star-forming galaxy. We first analyzed the dynamical evolution of the
supernova ejecta impacted by the jet. Then, we explored the parameter space
using simple prescriptions for the observed gamma-ray lightcurve. Finally, the
synchrotron and the inverse Compton spectral energy distributions for two types
of sources, a radio galaxy and a powerful blazar, are computed. For a radio
galaxy, the interaction between a supernova and a jet of power $\sim
10^{43}-10^{44}$~erg~s$^{-1}$ can produce apparent gamma-ray luminosities of
$\sim 10^{42}-10^{43}$~erg~s$^{-1}$, with an event duty cycle of supernova
remnant (SNR) interacting with the jet close to one for one galaxy. For a
blazar with a powerful jet of $\sim 10^{46}$~erg~s$^{-1}$, the jet-supernova
ejecta interaction could produce apparent gamma-ray luminosities of $\sim
10^{43}-10^{44}$~erg~s$^{-1}$, but with a much lower duty cycle. The
interaction of supernovae with misaligned jets of moderate power can be
relatively frequent, and can result in steady gamma-ray emission potentially
detectable for sources in the local universe. For powerful blazars much farther
away, the emission would be steady as well, and it might be detectable under
very efficient acceleration, but the events would be rather infrequent.
