A genus formula for the positive \'{e}tale wild kernel

Let $F$ be a number field and let $i\ge 2$ be an integer. In this paper, we study the positive '{e}tale wild kernel ${\mathrm{W}\mathrm{K}}_{2i-2}^{\acute{et},+}F$, which is the twisted analogue of the $2$-primary part of the narrow class group. If $E\mathrm{/}F$ is a Galois extension of number fields with Galois group $G$, we prove a genus formula relating the order of the groups $({\mathrm{W}\mathrm{K}}_{2i-2}^{\acute{et},+}E{)}_G$ and ${\mathrm{W}\mathrm{K}}_{2i-2}^{\acute{et},+}F$.