  The main goal of the paper is to investigate theoretically the experimentally
observed fragmentation of the isovector $M1$ resonance in $^{208}$Pb within a
self-consistent model based on an energy-density functional (EDF) of the Skyrme
type. This fragmentation (spread of the $M1$ strength) is not reproduced in a
conventional one-particle--one-hole ($1p1h$) random-phase approximation (RPA)
and thus has to be investigated in the framework of more complicated models.
However, previously applied models of this type were not self-consistent. In
the present work, we use a recently developed renormalized version of the
self-consistent time blocking approximation (RenTBA) in which the
$1p1h\otimes$phonon configurations are included on top of the RPA $1p1h$
configurations. We have determined several sets of the parameters of the
modified Skyrme EDF fitted within the RenTBA and RPA and have found the
necessary condition of producing the fragmentation of the $M1$ resonance in
$^{208}$Pb in our model. We present also the results of the RenTBA and RPA
calculations for the first excited states of the natural parity modes in
$^{208}$Pb obtained with these modified parametrizations.
