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Does anybody know if the battery grip off my A7 iii will fit the A7 iv?
No, ILCE-A7M4 need VG-C4EM, your is VG-C3EM:
http://support.d-imaging.sony.co.jp/www/cscs/accessories/compatibility.php?area=gb&lang=en&cs_ref=sl...
Exactly, a VG-C4EM is needed, yours will not fit.