Isomorphism between symplectic and projective symplectic group in characteristic two

Statement

Suppose $K$ is a field and $m$ is a positive integer. The following are equivalent:

1. $K$ has characteristic two.
2. The symplectic group $Sp(2m,K)$ is a centerless group.
3. The natural quotient map from the symplectic group $Sp(2m,K)$ to the projective symplectic group $PSp(2m,K)$ is an isomorphism of groups.
4. The symplectic group $Sp(2m,K)$ and projective symplectic group $PSp(2m,K)$ are isomorphic groups.

In any other characteristic, the center of $Sp(2m,K)$ is cyclic of order two.