Make morphism from GF(p^2)'s multiplicative group to GF(p)'s multiplicative group

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Using

sage: F=GF(11)
sage: F.unit_group()
Multiplicative Abelian group isomorphic to C10

@FrédéricC: This works only for prime fields. E.g., GF(11^2).unit_group()errors out. Relevant bugreport: https://trac.sagemath.org/ticket/7234

This may be a replacement for .unit_group():

K = GF(11^2)
from sage.monoids.automatic_semigroup import AutomaticSemigroup
M = AutomaticSemigroup( [K.multiplicative_generator()], one=K.one(), category=Monoids().Finite().Subobjects() & Groups() )

Thank you very much for your answer. I managed to build K, but unfortunately, I was not able to build a morphism from it. The hom method seems to work differently from what I know for this class and throws errors I could not comprehend. Even K.hom([K.gens()[0]^2]) which should be the endomorphism $x$ gives $x^2$ does not work...

It is fairly easy to define a pythonic map $x\to x^{p+1}$. Yes, we completely ignore the mathematical structure. But to use it, and to be able to get such an involved instance knowing everything about structure - not only the map, but also domain, codomain, their properties... well, the hom and the instances we are starting with are not prepared for this particular usage... I would understand the need for a hom-instance implementing $\phi$, if this homomorphism would be (pre)composed with some other one, but also in that case, it is still too much yoga of arranging classes to play the game. If the particular set-theoretical map is enough for the programming purpose, i would go for it.