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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...

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...