  This paper aims at developing model-theoretic tools to study interpretable
fields and definably amenable groups, mainly in $\mathrm{NIP}$ or
$\mathrm{NTP_2}$ settings. An abstract theorem constructing definable group
homomorphisms from generic data is proved. It relies heavily on a stabilizer
theorem of Montenegro, Onshuus and Simon. The main application is a structure
theorem for definably amenable groups that are interpretable in algebraically
bounded perfect $\mathrm{NTP_2}$ fields with bounded Galois group (under some
mild assumption on the imaginaries involved), or in algebraically bounded
theories of (differential) NIP fields. These imply a classification of the
fields interpretable in differentially closed valued fields, and structure
theorems for fields interpretable in finitely ramified henselian valued fields
of characteristic $0$, or in NIP algebraically bounded differential fields.
