Conversation so far:
Did they have any clues?
probably FSB) are known to have targeted the webmail account of the murdered Russian journalist Anna Politkovskaya.
How did they target her email?
On 5 December 2005, RFIS initiated an attack against the account annapolitovskaya@US Provider1, by deploying malicious software.
Did they get into trouble for that?
CANNOTANSWER.

Question: Did they have any murder suspects?
Self-contained rewrite: Did FSB have any Anna Politkovskaya's murder suspects?