= Solution
The three order-four <subgroups> are
$$
\boxed{G_1=\{W,X,\widehat W,\widehat X\}=\langle X\rangle,\quad
G_2=\{W,Y,\widehat W,\widehat Y\}=\langle Y\rangle,\quad
G_3=\{W,Z,\widehat W,\widehat Z\}=\langle Z\rangle.}
$$
In each case the generator has powers $A,A^2=\widehat W,A^3=\widehat A,A^4=W$, so all three are cyclic and mutually isomorphic to $C_4$, a <cyclic group of order four>. For $G_1$ the table is
$$
\boxed{\begin{array}{c|cccc}
&W&X&\widehat W&\widehat X\\\hline
W&W&X&\widehat W&\widehat X\\
X&X&\widehat W&\widehat X&W\\
\widehat W&\widehat W&\widehat X&W&X\\
\widehat X&\widehat X&W&X&\widehat W
\end{array}}
$$
The maps $X^j\mapsto Y^j$ and $X^j\mapsto Z^j$ are explicit <group isomorphisms>. These are all order-four <subgroups>: any such <subgroup> contains an element other than $W,\widehat W$, and that element has order four and already generates the <subgroup>. They share the order-two <subgroup> $\{W,\widehat W\}$ and are each normal, being of index two.
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