This is a long comment rather than an answer.

It provides some code to define the objects in the question,
in the hope it can help answer it.

### Galois subgroups and their fixed fields

Starting with Sage 9.4.beta1, in which the following ticket was merged:

one can get subgroups of the Galois group of a number field, and
the associated fixed fields.

Define a cyclotomic field:

```
sage: K.<z> = CyclotomicField(5)
```

Get its Galois group:

```
sage: G = K.galois_group()
sage: G
Galois group 4T1 (4) with order 4 of x^4 + x^3 + x^2 + x + 1
```

Get that group's subgroups:

```
sage: G.subgroups()
[Subgroup generated by [()] of
(Galois group 4T1 (4) with order 4 of x^4 + x^3 + x^2 + x + 1),
Subgroup generated by [(1,3)(2,4)] of
(Galois group 4T1 (4) with order 4 of x^4 + x^3 + x^2 + x + 1),
Subgroup generated by [(1,2,3,4), (1,3)(2,4)] of
(Galois group 4T1 (4) with order 4 of x^4 + x^3 + x^2 + x + 1)]
```

Get the associated fixed fields:

```
sage: H, I, J = G.subgroups()
sage: H.fixed_field()
(Cyclotomic Field of order 5 and degree 4,
Identity endomorphism of Cyclotomic Field of order 5 and degree 4)
sage: I.fixed_field()
(Number Field in z0 with defining polynomial x^2 - x - 1 with z0 = -0.618033988749895?,
Ring morphism:
From: Number Field in z0 with defining polynomial x^2 - x - 1 with z0 = -0.618033988749895?
To: Cyclotomic Field of order 5 and degree 4
Defn: z0 |--> z^3 + z^2 + 1)
sage: J.fixed_field()
(Rational Field,
Coercion map:
From: Rational Field
To: Cyclotomic Field of order 5 and degree 4)
```

The action of Galois group elements on field elements is as follows:

```
sage: G.gens()
[(1,2,3,4)]
sage: g = G.gen()
sage: g
(1,2,3,4)
sage: g.order()
4
sage: g(z)
z^2
sage: g(z^2)
-z^3 - z^2 - z - 1
sage: (g^2)(z)
-z^3 - z^2 - z - 1
```

### Multiplicative subgroups of cyclic rings

Now let us consider the the cyclic ring on 5 elements:

```
sage: C = Zmod(5)
sage: C
Ring of integers modulo 5
```

There are two ways to get the subgroups of its group of units.

The method `multiplicative_subgroups`

gives tuples of generators
for these subgroups.

```
sage: C.multiplicative_subgroups()
((2,), (4,), ())
```

The other way is to construct the ring's unit group and get its subgroups.

```
sage: U = C.unit_group()
sage: U.subgroups()
[Multiplicative Abelian subgroup isomorphic to C4 generated by {f},
Multiplicative Abelian subgroup isomorphic to C2 generated by {f^2},
Trivial Abelian subgroup]
```

There was recently a discussion about working with these subgroups at:

### Putting the pieces together

Hopefully someone sees how to put these pieces together
to answer the question satisfactorily!