Hi,
I think 2 and 3 in L2M2M3 represent the subshells (not shell). So it is not highly significant to draw/show these transitions in schematic drawings or in shells schemes (i guess shell schemes means you draw a circular nucleus and draw K,L,M,etc circles around it!). Otherwise one can draw spectroscopic notations (here you draw K, L, M, lines instead of circles).
For example L2M2M3 means: L2 electron is knocked/removed and this vacancy is filled by M2 electron. After wards M3 electron (generally Auger electron) will be automatically removed.
In this article you can find some relevant information on LMM Auger emission.
DOI: 10.1007/s10751-005-9177-1
Hope it helps.
Cheers