To describe the transfer targets and kernels of a quantum class group G_{p}^{2}(K),
it is necessary to investigate certain quotients G of coclass cc(G) = 1.
First, we consider abelianizations of diamond type (p,p) for an odd prime p ≥ 3.
The proofs of the following two theorems are due to
Heider and Schmithals.
However, here we express the statements in a purely group theoretical form
which is convenient to emphasize our intention.

1.1. Coarse Transfer Kernel Type (cTKT)
Theorem 1.
Let p ≥ 3 be an odd prime, n ≥ 2 an integer,
and G ∈ CF(n,n;p) a pgroup of order G = p^{n} and class cl(G) = n1,
having an abelian maximal subgroup A.
(Consequently, G is of coclass cc(G) = 1
with abelianization G/G' of diamond type (p,p)
and cyclic lower central factors (G_{j}:G_{j+1}) = p for 2 ≤ j ≤ n1.)
Denote the transfer from G to A by T : G/G' → A. Then
1.1. ker(T) = G/G', if and only if G ≅ G_{0}^{n}(0,0),
1.2. ker(T) = A/G', if and only if G ≅ G_{0}^{n}(0,1),
2. ker(T) = M/G', for some maximal subgroup M ≠ A, if and only if G ≅ G_{0}^{n}(g^{i},0),
where g ≥ 1 denotes the smallest primitive root modulo p and 0 ≤ i ≤ gcd(n2,p1)1.
Remark 1.
In case 1.1, T is called a total transfer,
in case 1.2 a partial transfer with fixed point, and
in case 2 a partial transfer without fixed point.
The coarse Taussky type (A) of T includes cases 1.1 and 1.2, characterised by ker(T) ∩ A/G' > 1,
whereas Taussky type (B) coincides with case 2, where ker(T) ∩ A/G' = 1.
Blackburn's metabelian pgroups G_{a}^{n}(z,w) are defined in our paper
Transfers of metabelian pgroups.

1.2. Transfer Target Type (TTT)
Theorem 2.
Let p ≥ 3 be an odd prime, n ≥ 2 an integer,
and G ∈ CF(n,n;p) a pgroup of order G = p^{n} and class cl(G) = n1,
having an abelian maximal subgroup A of index (G:A) = p and order A = p^{m}, m = n1.
Denote the transfer from G to A by T : G/G' → A.
Then the structure of A is given by
1.1. A ≅ ((p^{q+1},…,p^{q+1})_{r times},(p^{q},…,p^{q})_{(p1)r times}),
where m = q(p1)+r, q ≥ 1, 0 ≤ r < p1, if m ≥ p+1,
1.2. A ≅ (p^{2},(p,…,p)_{p2 times}), if m = p and T is of type (A),
1.3. A ≅ (p^{2},(p,…,p)_{p2 times}), if m = p, T is of type (B), and G not ≅ G_{0}^{p+1}(1,0),
2.1. A ≅ (p,…,p)_{p times}, if m = p, T is of type (B), and G ≅ G_{0}^{p+1}(1,0),
2.2. A ≅ (p,…,p)_{m times}, if 1 ≤ m ≤ p1 and T is of type (A),
3. A ≅ (p^{2},(p,…,p)_{m2 times}), if 2 ≤ m ≤ p1 and T is of type (B).
Remark 2.
In cases 1.1, 1.2, 1.3, A is a nearly homocyclic abelian pgroup of prank p1,
in cases 2.1, 2.2 an elementary abelian pgroup of prank ≤ p,
and in case 3, A has an exceptional structure of prank ≤ p2.
The only case with elevated prank p is 2.1, where G ≅ G_{0}^{p+1}(1,0) ≅ Syl_{p}A(p^{2}).

