  The electronic structure, magnetism and Fermi surface of a hole self-doped
potential iron-based superconductor KCa$_2$Fe$_4$As$_4$O$_2$ (12442) were
studied by a first-principes DFT/GGA approach. The projection onto Wannier
functions basis is proposed to calculate directly the self-doping value. In
particular for KCa$_2$Fe$_4$As$_4$O$_2$ it is 0.75 hole on Fe-3d states.
Surprisingly, for the KCa$_2$Fe$_4$As$_4$O$_2$ it was found that except the
Fe-3d bands the O-2p bands also cross the Fermi level, which is quite anomalous
behavior for O-2p states. In general, the O-2p states are usually fully
occupied in transition metal oxides. Here it is not the case due to the fact
that CaO layer gives some of the electrons to the FeAs layer, thus leaving O-2p
states partially occupied. The O-2p states form additional Fermi surface sheet
with propeller-like shape around $\Gamma$-point and modify the Fe-3d Fermi
surface sheets due to the hybridization. To prove the presence of anomalous
metallic O-2p band in KCa$_2$Fe$_4$As$_4$O$_2$ the comparative study with the
same family systems KCa$_2$Fe$_4$As$_4$F$_2$ and RbGd$_2$Fe$_4$As$_4$O$_2$
(where it is not the case) was performed. The magnetic ground state is found to
be antiferromagnetic with checkerboard ordering. Therefore, we expect that the
KCa$_2$Fe$_4$As$_4$O$_2$ might be a potential superconductor with unusual
properties.
