Synthesists are a unique summoner archetype that combines with the eidolon, which can range from a Hulk or werewolf relationship to powered-up final forms, to a super-suit. The eidolon is a separate creature by the rules, but the summoner wears the eidolon.
A synthesist eidolon does not get its own feats, but the hybrid gets to use the feats of the summoner generally. Eidolons do have a limit on the number of natural attacks they may have. My question is how a synthesist would handle having a natural attack from race, feat, or class feature? For the sake of an example, Eldritch Heritage with the Serpentine Bloodline:
Serpent’s Fang (Ex): At 1st level, you can grow fangs as a free action. These fangs are treated as a natural weapon inflicting 1d4 points of damage plus your Strength modifier (1d3 if you are Small) plus poison (Bite—injury; save Fort DC 10 + 1/2 your sorcerer level + your Constitution modifier; frequency 1/round for 6 rounds; effect 1 Con damage; cure 1 save).
This seems like a good test as the feat gives the ability to grow the fangs. A second scenario to consider would be a race with natural weapons. Would these natural attacks of the summoner be:
A) Included as part of the eidolon and so counted against the limit.
B) Not accessible as they are natural attacks of the summoner rather than the eidolon.
C) Available to the eidolon, but not considered for the limit.
Ultimately such a case is up to the DM of a game and such oddities are a contributing factor for why the synthesist has always been banned for PFS, but I have always been a firm believer in trying to understand something like this before springing it on a DM. Thematic-wise, I would think racial natural attacks that always exist would fit under B, while the ability to grow them would fit under A or C depending on whether ruling for balance or thematics respectively, but I may be missing a rule along the way.