  We present the explicit form of the next-to-next-to-leading order (N$^2$LO)
Skyrme interaction in momentum space by including the fourth-order gradient
potentials to the standard Skyrme interaction. With the N$^2$LO Skyrme
interaction, we evaluate the second-order corrections to the nuclear bulk
quantities of nuclear matter: equation of state (EoS) of isospin symmetric and
pure neutron matter, density-dependent in-medium effective nucleon mass,
isospin-asymmetry energy, pressure and incompressibility. These second-order
contributions are ultraviolet (UV) divergent due to the zero range character of
the interaction and renormalized using the techniques of dimensional
regularization (DR) with the minimal subtraction scheme (MS). We adjust the 18
parameters of the interaction by performing a global fit to the nuclear bulk
quantities. Besides the too strong dependence $k_F^{12}$ of several
second-order corrections, a very good reproduction of a realistic nuclear
matter saturation curve with all the nuclear bulk quantities in the density
region $0.12<\rho<0.20\;$ fm$^{-3}$ is obtained.
