  It is important to develop a unified theoretical framework to describe the
nuclear experiments and astrophysical observations based on the same effective
nuclear interactions. Based on the so-called Skyrme pseudopotential up to
next-to-next-to-next-to-leading order, we construct a series of extended Skyrme
interactions by modifying the density-dependent term and fitting the empirical
nucleon optical potential up to above $1$ GeV, the empirical properties of
isospin symmetric nuclear matter, the microscopic calculations of pure neutron
matter and the properties of neutron stars from astrophysical observations. The
modification of the density-dependent term in the extended Skyrme interactions
follows the idea of Fermi momentum expansion and this leads to a highly
flexible density behavior of the symmetry energy. In particular, the values of
the density slope parameter $L$ of the symmetry energy for the new extended
Skyrme interactions range from $L = -5$ MeV to $L = 125$ MeV by construction,
to cover the large uncertainty of the density dependence of the symmetry
energy. Furthermore, in order to consider the effects of isoscalar and
isovector nucleon effective masses, we adjust the momentum dependency of the
single-nucleon optical potential and the symmetry potential of these new
extended Skyrme interactions and construct a parameter set family, by which we
systematically study the impacts of the symmetry energy and the nucleon
effective masses on the properties of nuclear matter and neutron stars. The new
extended Skyrme interactions constructed in the present work will be useful to
determine the equation of state of isospin asymmetric nuclear matter,
especially the symmetry energy, as well as the nucleon effective masses and
their isospin splitting, in transport model simulations for heavy-ion
collisions, nuclear structure calculations and neutron star studies.
