Working with large finite fields of characteristic 2; how to coerce them?

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First of all that in does not translate (mathematically and/or pythonically) as an inclusion. The following works:

p, r, s, rs = 2, 16, 16, 256
Fp       = GF(p)
Fr.<a>  = GF(r)
Fs.<b>  = GF(s)
Frs.<c> = GF(rs)

Now we have some fields and ask for existent coercion maps.

sage: Fr.has_coerce_map_from(Fp)
True
sage: Fs.has_coerce_map_from(Fp)
True
sage: Frs.has_coerce_map_from(Fp)
True
sage: Frs.has_coerce_map_from(Fr)
False
sage: Frs.has_coerce_map_from(Fs)
False

Why False when false? Because for instance a is an element inside $\Bbb F_{16}=\Bbb F_2(a)$ and we do not have a canonical choice of it inside the field $\Bbb F_{256}$ realized as $\Bbb F_2(c)$. Explicitly, every root of the minimal polynomial of $a$ from the last field gives rise to a possible injection between fields, $\Bbb F_2(a)\to \Bbb F_2(c)$:

sage: a.minimal_polynomial().roots(ring=Frs)
[(c^6 + c^3 + c^2 + c, 1),
 (c^6 + c^3 + c^2 + c + 1, 1),
 (c^7 + c^4 + c^3, 1),
 (c^7 + c^4 + c^3 + 1, 1)]

So which coercion should sage or a friend chose for us?

Note: Posting as an Anonymous is possible, but the dialog suffers if there is a follow up...

For instance, which is here the question? How to coerce? Is it enough to work in the bigger field? (Thus defining $a$ as one of the root values above, and $b$ maybe an other one... Who knows?!)

Look like it works only with finite field omiting variable name.

sage: p=2
sage: r=16
sage: s=16
sage: F_p=GF(p)
sage: F_r=GF(p**r);F_r
Finite Field in z16 of size 2^16
sage: F_s=GF(p**(r*s));F_s
Finite Field in z256 of size 2^256
sage: F_p in F_r
False
sage: F_p in F_s
False
sage: F_r in F_s
False
sage: a,=F_r.gens()
sage: b,=F_s.gens()
sage: a+b
z256^253 + z256^251 + z256^248 + z256^246 + z256^243 + z256^239 + z256^236 + z256^235 + z256^234 + z256^233 + z256^231 + z256^229 + z256^226 + z256^225 + z256^223 + z256^222 + z256^220 + z256^218 + z256^215 + z256^213 + z256^211 + z256^210 + z256^209 + z256^206 + z256^205 + z256^203 + z256^202 + z256^199 + z256^195 + z256^192 + z256^191 + z256^190 + z256^185 + z256^182 + z256^180 + z256^179 + z256^176 + z256^175 + z256^172 + z256^170 + z256^169 + z256^167 + z256^165 + z256^163 + z256^162 + z256^161 + z256^160 + z256^156 + z256^155 + z256^153 + z256^151 + z256^150 + z256^149 + z256^148 + z256^145 + z256^144 + z256^143 + z256^141 + z256^139 + z256^138 + z256^135 + z256^133 + z256^131 + z256^127 + z256^126 + z256^123 + z256^122 + z256^120 + z256^119 + z256^118 + z256^116 + z256^114 + z256^113 + z256^110 + z256^107 + z256^105 + z256^101 + z256^99 + z256^96 + z256^94 + z256^92 + z256^91 + z256^88 + z256^84 + z256^79 + z256^78 + z256^75 + z256^73 + z256^72 + z256^68 + z256^66 + z256^64 + z256^61 + z256^59 + z256^58 + z256^56 + z256^54 + z256^52 + z256^50 + z256^46 + z256^43 + z256^42 + z256^40 + z256^36 + z256^35 + z256^33 + z256^31 + z256^29 + z256^22 + z256^21 + z256^20 + z256^19 + z256^14 + z256^9 + z256^8 + z256^7 + z256^6 + z256^4 + z256^2