update sql query with case statement

I have the following select query

select
w.login,
w.RolesSetThru,
e.Authenticity as eAuthenticity,
a.Authenticity as aAuthenticity ,
case when e.Authenticity in (1,3) then 'AD'
     when a.Authenticity in (1,3) then 'AD'
else
      'ERMXref'
end            as Authenticity  
from  controls..WrkUser w
left outer join controls..WrkERMGroupMember e on
 w.Login = e.Login
left outer join controls..WrkADGroupMember a on
 w.Login = a.Login

I want to update w.RolesSetThru column with the values that I get from case statement. Can someone please show me how I can get that done