  Binary stellar evolution has been studied as important pathway to initiate
various transient events like supernovae (SNe). Although the common envelope
(CE) in a binary, outcomes of the CE and conditions for the SN explosion during
the CE phase are uncertain, it has been suggested that SN explosions can be
triggered during the CE phase. In this work, we explore formation and evolution
routes of carbon/oxygen (CO) WDs binaries in order to investigate mergers of CO
WDs and cores of non-degenerate stars during the CE phase as possible origins
for SNe under the core merger detonation (CMD) scenario by considering several
binary physical models. Evolution of CO WD + intermediate mass normal
(non-degenerate and hydrogen-rich) star binaries lead to mergers during the CE
phases still may trigger type Ia SNe (SNe Ia) interacting with circumstellar
material under different models. Mergers between CO WDs and cores of He
(non-degenerate and helium-rich) stars within the CE phases are rare comparing
to other CE merger events. These two channels may produce peculiar SNe Ia such
as over-luminous/super-chandrasekhar mass SNe Ia as certain fraction of them
have combined core masses $\geq2M_\odot$. In the channel of CO WD + massive
star ($\geq8M_\odot$) CE events, we find that rates of mergers between CO WDs
and He cores of massive stars are (0.85 - 12.18)$\times10^{-5}{M_\odot}^{-1}$
which may initiate type II superluminous SNe like SN 2006gy, and delay times
from this scenario are in the range of 34 - 120 Myr. Our results based on the
CMD model are comparable with observational results of peculiar SNe.
