A group of order 20

Suppose, by way of contradiction, that there is a normal $2$-Sylow subgroup $D$.

If $D$ is cyclic, then it has exactly one involution, and this would form then a conjugacy class made of one element, which does not show up in the class equation. (The "$1$" in the class equation is already taken by the identity.)

This argument actually shows that there is no normal subgroup of order $2$.

If $D$ is a a Klein four-group, it has exactly three involutions, which would then yield a conjugacy class of order $1$ or $3$, again not there.