This question was the motivation for two papers of mine with John R. Britnell. In Commuting conjugacy classes, an application of Hall's Marriage Theorem to group theory, J. Group. Th, 12 (2009) 795–802, we showed that there is a bijection between the non-split classes in the kernel and those not in the kernel, such that if $C \subseteq \ker \phi$ is paired with $D \subseteq G \backslash \ker \phi$ then $C$ and $D$ have representatives that commute. Here a class $C$ is non-split if its centralizer is not contained in $\ker \phi$; clearly this is a necessary condition for the claimed bijection. This immediately implies that there are at least as many conjugacy classes in $\ker \phi$ as conjugacy classes in $G \backslash \ker \phi$.

The proof is elementary, by counting the number of conjugacy classes that can be paired with each $C$, and showing that this number satisfies the conditions for Hall's Marriage Theorem.

In the case of the symmetric group this implies the identity counting partitions into an odd/even number of even parts mentioned in the question. (But while it says a bijection exists, it does not give one explicitly.) In Combinatorial proof of a theorem on partitions into an even or odd number of parts, J. Comb. Th. Ser A. 21 (1976) 100–103, Gupta gave an explicit bijective proof of this identity that happens to have the commuting property: see Section 3.1 of my joint paper.

In On the distribution of conjugacy classes between the cosets of a group in a cyclic extension, Bull. Lond. Math. Soc. 40 (2008) 897–906, we generalised the counting part of this result to arbitrary cyclic quotients.

This question was the motivation for two papers of mine with John R. Britnell. In Commuting conjugacy classes, an application of Hall's Marriage Theorem to group theory, J. Group. Th, 12 (2009) 795–802, we showed that there is a bijection between the non-split classes in the kernel and those not in the kernel, such that if $C \subseteq \ker \phi$ is paired with $D \subseteq G \setminus \ker \phi$ then $C$ and $D$ have representatives that commute. Here a class $C$ is non-split if its centralizer is not contained in $\ker \phi$; clearly this is a necessary condition for the claimed bijection. This immediately implies that there are at least as many conjugacy classes in $\ker \phi$ as conjugacy classes in $G \setminus \ker \phi$.

The proof is elementary, by counting the number of conjugacy classes that can be paired with each $C$, and showing that this number satisfies the conditions for Hall's Marriage Theorem.

In the case of the symmetric group this implies the identity counting partitions into an odd/even number of even parts mentioned in the question. (But while it says a bijection exists, it does not give one explicitly.) In Combinatorial proof of a theorem on partitions into an even or odd number of parts, J. Comb. Th. Ser A. 21 (1976) 100–103, Gupta gave an explicit bijective proof of this identity that happens to have the commuting property: see Section 3.1 of my joint paper.

In On the distribution of conjugacy classes between the cosets of a group in a cyclic extension, Bull. Lond. Math. Soc. 40 (2008) 897–906, we generalised the counting part of this result to arbitrary cyclic quotients.

This question was the motivation for two papers of mine with John R. Britnell. In Commuting conjugacy classes, an application of Hall's Marriage Theorem to group theory, J. Group. Th, 12 (2009) 795–802, we showed that there is a bijection between the non-split classes in the kernel and those not in the kernel, such that if $C \subseteq \ker \phi$ is paired with $D \subseteq G \backslash \ker \phi$ then $C$ and $D$ have representatives that commute. Here a class $C$ is non-split if its centralizer is not contained in $\ker \phi$; clearly this is a necessary condition for the claimed bijection. This immediately implies that there are at least as many conjugacy classes in $\ker \phi$ as conjugacy classes in $G \backslash \ker \phi$.

In the case of the symmetric group this implies the identity counting partitions into an odd/even number of even parts mentioned in the question. (But while it says a bijection exists, it does not give one explicitly.) In Combinatorial proof of a theorem on partitions into an even or odd number of parts, J. Comb. Th. Ser A. 21 (1976) 100–103, Gupta gave an explicit bijective proof of this identity that happens to have the commuting property: see Section 3.1 of my joint paper.

In On the distribution of conjugacy classes between the cosets of a group in a cyclic extension, Bull. Lond. Math. Soc. 40 (2008) 897–906, we generalised the counting part of this result to arbitrary cyclic quotients.

This question was the motivation for two papers of mine with John R. Britnell. In Commuting conjugacy classes, an application of Hall's Marriage Theorem to group theory, J. Group. Th, 12 (2009) 795–802, we showed that there is a bijection between the non-split classes in the kernel and those not in the kernel, such that if $C \subseteq \ker \phi$ is paired with $D \subseteq G \backslash \ker \phi$ then $C$ and $D$ have representatives that commute. Here a class $C$ is non-split if its centralizer is not contained in $\ker \phi$; clearly this is a necessary condition for the claimed bijection. This immediately implies that there are at least as many conjugacy classes in $\ker \phi$ as conjugacy classes in $G \backslash \ker \phi$.

The proof is elementary, by counting the number of conjugacy classes that can be paired with each $C$, and showing that this number satisfies the conditions for Hall's Marriage Theorem.

In the case of the symmetric group this implies the identity counting partitions into an odd/even number of even parts mentioned in the question. (But while it says a bijection exists, it does not give one explicitly.) In Combinatorial proof of a theorem on partitions into an even or odd number of parts, J. Comb. Th. Ser A. 21 (1976) 100–103, Gupta gave an explicit bijective proof of this identity that happens to have the commuting property: see Section 3.1 of my joint paper.

In On the distribution of conjugacy classes between the cosets of a group in a cyclic extension, Bull. Lond. Math. Soc. 40 (2008) 897–906, we generalised the counting part of this result to arbitrary cyclic quotients.