Administer - administrate
The verb in English is traditionally to administer.
From this, we derive the noun administration. (This is perfectly regularly formed.) The first appearance of to administrate, a back-formation from the noun, is recorded in OED as 1639. It has increased in frequency from the late 19th century onwards.
Administrate seems unnecessary; and to AWE and many academics ugly as well. Don't use it. Use the traditional administer instead. (An exception may be made, perhaps, when administrate is used in the sense 'be an administrator of' - he administrates the University, i.e. he is a P.V.C. or other senior person.)