TITLE: Are groups completely determined by their representations?
QUESTION [7 upvotes]: Recently, I became interested in representation theory, and I found out a natural philosophical (vague) question:  are groups completely determined by their representations?
To be specific, I want to know answers about following questions: 
(1) For any (discrete) group $G$ and a field $k$, let $\text{Rep}_{G,k}$ be the category of representations of $G$ over $k$. If $\text{Rep}_{G,k}$ and $\text{Rep}_{G',k}$ are equivalent for all $k$, does this implies $G\simeq G'$? How about finite dimensional representation? 
(2) What happens if '(discrete) group' changed to other kinds of groups (and representation also changed by appropriate alternatives), such as continuous representations of topological groups, or smooth representations of Lie groups?
Since (1) and (2) are just my formulations of the original question, please let me know if there exist other better-formulated specific questions related to the original question. Also, if you know related theorems (or conjectures), even if they do not exactly focus on (1) and (2), please tell me. Thanks in advance.

REPLY [5 votes]: Take a look at the Annals paper "A Counterexample to the Isomorphism Problem for Integral Group Rings" by Martin Hertweck. He constructs two finite groups $G$, $H$ such that the integral group rings $\mathbb{Z}G$ and $\mathbb{Z}H$ are isomorphic as rings.  In particular this means that $Rep_{G,R}$ and $Rep_{H,R}$ equivalent as categories for any base ring $R$.
Of course categories of representations of groups come equipped with additional structure (i.e. a symmetric monoidal structure), and these equivalences will not preserve those structures.