WordNet sense 1 (give an assignment to (a person) to a post, or assign a task to (a person)):

delegate, depute, designate, delegate

WordNet sense 2 (make undue claims to having):

arrogate

WordNet sense 3 (give out or allot):

allot, portion

WordNet sense 4 (attribute or give):

put

WordNet sense 5 (attribute or credit to):

ascribe, impute, attribute, attribute

WordNet sense 6 (select something or someone for a specific purpose):

specify