Administer is the verb form for administration or administrator.
The word administrate is an incorrect form of the verb created by some who drop the -ion suffix of administration.
Incorrect: He did a great job of administrating the estate.
Correct: He did a great job of administering the estate.
Be careful when forming verbs from nouns that end in -ation, as the correct verb form may not end in -ate.
See also Orient or Orientate?