German orthography makes a difference between ß and ss, to be over-simplifying, ß follows a long vowel and diphthongs, which are considered long vowels (au is a diphthong), ss follows short vowels. Since 1996 it is permissible to use ss instead of ß when a typeface does not contain ß. The exception is names (see Theodor Heuss, the first post-WWII president of West Germany), so whether to use ß or ss for your family member depends on whether the family historically used to use ß. If that is the case her name should be Außem.