DRIVEN is related to EIDRVN in the same way as BEGUMF is related to:

In DRIVEN the letters occupy positions 1-6 as D-R-I-V-E-N; the output EIDRVN rearranges them by picking original positions in the order 5,3,1,2,4,6. Applying the same order to BEGUMF (B1-E2-G3-U4-M5-F6) gives M-G-B-E-U-F, that is MGBEUF.