The partitions on \\(T\\) are

[1234],[12][34],[123][4],[124][3],[12][3][4].

The images of these partitions under \\(g^*\\) are

[1234],[12][34],[123][4],[124][3],[12][3][4]

where now \\(1\\) and \\(2\\) are distinct elements of \\(S\\), though they always appear in the pulled-back partitions together (because \\(g\\) maps them to the same element of \\(T\\)).