Classification of Degenerations and Picard Lattices of Kählerian K3 Surfaces with Symplectic Automorphism Group $C_4$

In the author's papers of 2013–2018, the degenerations and Picard lattices of Kählerian K3 surfaces with finite symplectic automorphism groups of high order were classified. For the remaining groups of small order—$D_6$, $C_4$, $(C_2)^2$, $C_3$, $C_2$, and $C_1$—the classification was not completed because each of these cases requires very long and difficult considerations and calculations. The case of $D_6$ was recently completely studied in the author's paper of 2019. In the present paper an analogous complete classification is presented for the cyclic group $C_4$ of order $4$.