Afghanistan has had puppet leaders before, rulers who served only the interests of foreign occupiers, whether British or Soviet.
And I don't think we can assume that she was some kind of Mitt puppet, since she's now been running the firm for two decades.
Set up a puppet government; then use that regime to do your bidding.
"doll moved by strings or wires" (later applied to puppets in glove form), 1530s, later form of Middle English popet "doll" (c.1300; cf. poppet), from Old French popette "little doll, puppet," diminutive of popee "doll, puppet" (13c., Modern French poupée), from Vulgar Latin *puppa, from Latin pupa "girl; doll" (see pupil (n.1)).
Metaphoric extension to "one whose actions are manipulated by another" first recorded 1540s (as poppet). Puppet show attested from 1650s, earlier puppet-play (1550s). Puppet government is attested from 1884 (in reference to Egypt).