TITLE: Almost uniquely generated groups
QUESTION [6 upvotes]: This is inspired by this question.


Does there exist an infinite finitely generated group having 
(a) a unique
(b) finitely many
inclusion-minimal generating set(s) up to automorphisms?


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2.$ $ Which finite groups have a unique inclusion-minimal generating set up to automorphisms?

REPLY [3 votes]: Here is an honors thesis studying exactly this property for finite groups, which the author calls UMP, and all conclusions in the various answers are in the thesis. More examples are also given. A related paper by the same author is here