  We investigate an intermittent stochastic process, in which the diffusive
motion with time-dependent diffusion coefficient $D(t)\sim t^{\alpha-1}$,
$\alpha>0$ (scaled Brownian motion), is stochastically reset to its initial
position and starts anew. The resetting follows a renewal process with either
exponential or power-law distribution of the waiting times between successive
renewals. The resetting events, however, do not affect the time dependence of
the diffusion coefficient, so that the whole process appears to be a
non-renewal one. We discuss the mean squared displacement of a particle and
probability density function of its positions in such a process. We show that
scaled Brownian motion with resetting demonstrates a rich behavior whose
properties essentially depend on the interplay of the parameters of the
resetting process and the particle's displacement in a free motion. The motion
of particles can remain either almost unaffected by resetting, but can also get
slowed down or even be completely suppressed. Especially interesting are the
nonstationary situations in which the mean squared displacement stagnates but
the distribution of positions does not tend to any steady state. \color{black}
This behavior is compared to the situation (discussed in the other paper of
this series) in which the memory on the value of the diffusion coefficient at a
resetting time is erased, so that the whole process is a fully renewal one. We
show that the properties of the probability densities in such processes
(erazing or retaining the memory on the diffusion coefficient) are vastly
different. \color{black}
