= Solution
Use the leading lepton-orbital classification of a <beta decay transition>. For <lepton>-pair <orbital angular momentum> $l$, <Fermi transitions> have pair <spin> $S=0$ and total lepton <angular momentum> $J=l$; <Gamow-Teller transitions> have $S=1$ and $J=|l-1|,\ldots,l+1$. <Addition of angular momentum> requires
$$
|J_i-J_f|\leq J\leq J_i+J_f,
$$
while the nuclear <parity> change requires $\pi_f/\pi_i=(-1)^l$. The least allowed $l$ gives the order: $l=0$ means an <allowed beta decay>, and $l=n>0$ an $n$th-<forbidden beta decay>. Fermi-type and Gamow-Teller-type below refer to these spin-independent and spin-dependent contributions, including their higher-orbital generalizations.
* For $^{129}\mathrm I\to{}^{129}\mathrm{Xe}$, the nuclear <spin> changes by $2$ and <parity> is unchanged. An $l=0$ <lepton> pair cannot carry $J\geq2$; $l=1$ would have the wrong <parity>. At $l=2$, <Fermi transitions> can carry $J=2$, and <Gamow-Teller transitions> can carry $J=1,2,3$, including the permitted transfers $2$ and $3$. Therefore this is a \b[second-forbidden, nonunique transition, with both Fermi-type and Gamow-Teller-type contributions permitted]. The quantum numbers do not determine their actual relative matrix elements.
* For $^{73}\mathrm{Kr}\to{}^{73}\mathrm{Br}$, $\Delta J=1$ and <parity> is unchanged. Already $l=0,S=1$ carries $J=1$. The $l=0$ operator for a <Fermi transition> has rank zero and cannot connect these unequal nuclear <spins>. This is an \b[allowed Gamow-Teller transition]. The decrease of <atomic number> makes the nuclear conversion proton-to-neutron; the <angular momentum> classification applies to either the energetically available <beta-plus decay> or <electron capture> branch.
* For $^{158}\mathrm{Yb}\to{}^{158}\mathrm{Tm}$, $0^+\to2^-$ requires transfer $J=2$ and a <parity> change. The lowest possible orbital order is $l=1$. The $S=0$ pair for a <Fermi transition> then has only $J=1$, whereas the $S=1$ pair for a <Gamow-Teller transition> can attain $J=2$. Thus this is a \b[first-forbidden unique Gamow-Teller-type transition]. Here unique means the maximal transfer $J=l+1$ at the leading orbital order, as in a <unique forbidden beta decay>.
* For $^{22}\mathrm{Mg}\to{}^{22}\mathrm{Na}$, $0^+\to0^+$ permits the scalar operator for a <Fermi transition> with $l=0$. The allowed <Gamow-Teller transition> operator has rank one, and the triangle rule excludes $0\to0$. Interpreting the printed logarithm in the usual <comparative half-life> convention gives
$$
ft=10^{3.48}\,\mathrm s=3.02\times10^3\,\mathrm s.
$$
This small value supports the expected \b[superallowed Fermi transition] between <isobaric analog states>. The $0^+\to0^+$ quantum numbers alone would establish allowed Fermi character, but would not by themselves establish the large analog-state overlap implied by superallowed strength.
In the order listed, the concise classification is
$$
\boxed{\begin{array}{ll}
1:\ \mathrm{F+GT},\ \text{second forbidden};&2:\ \mathrm{GT},\ \text{allowed};\\
3:\ \mathrm{GT},\ \text{first forbidden unique};&4:\ \mathrm{F},\ \text{superallowed}.
\end{array}}
$$
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