There are two types of numerically trivial involutions of an Enriques surface according to their period lattice. One is of $U\left(2\right)\perp U\left(2\right)$-type and the other is of $\left(U\right(2\left)\perp U\right(2\left)\right)$-type. An Enriques surface with an involution of $\left(U\right(2\left)\perp U\right(2\left)\right)$-type is doubly covered by a Kummer surface of product type, and such involutions are classified again into two types according to the parity of the corresponding Göpel subgroups. Involutions of odd $\left(U\right(2\left)\perp U\right(2\left)\right)$-type are constructed from the standard Cremona involutions of the quadric surface and closely related with quartic del Pezzo surfaces.