Pick up the phone
...or better yet, see your manager in person.
First, if you turn this into a public fight, it will alienate your manager and reflect poorly on you both. It is normal and reasonable to want some sort of public vindication, but you can't let that short term goal undermine your long-term work environment*.
Second, you might be wrong. Stakeholders have all sorts of opinions. Many of them are good, many are...not. Generally, you can figure out where the most important direction is coming from by figuring out who signs your paychecks. If a stakeholder says "jump" and your manager says "don't", your feet should stay planted on the ground. Whether this is idiocy or top cover depends primarily on how well you and your manager communicate (and nothing says you can't also be an advocate for jumping, if jumping is the right answer!).
At the same time though, your tasking is often going to be nebulous things like "make the stakeholders happy" or "do what those guys say". In that case, you may be juggling competing demands. Worse, some of the demanders may go to your manager if they think that will get what they want. If your manager hears only one side of the story, he may have a skewed view based on that, and may go charging off to take care of it.
So, how do you fix these things? Communication. What medium is best for communication for situations like "why did you throw me under the bus, jerk?!" depends on your goals. If you want to fix it, make it as personal as possible. Face to face and in private is best. Next best is skype or voice. I assume this is what you want. (If you want to document it for a grievance or to file a lawsuit or something, email or snail-mail is best. But I assume you don't want that.)
So, what do you do specifically? Call your boss. Ask whether this is a good time to talk for ten minutes (or as long as think this will take). Then talk about your feelings with a lot of "I" statements†. Roughly:
Hey Boss, I have a problem. Bob and Fred said to frobulate the juicer
clockwise, but Sue strongly objected and argued for counterclockwise
frobulation. They took the discussion offline, and I have been waiting
for them to get back to me with frobulation requirements. I think
[clockwise/counterclockwise/waiting for them to hash it out] is best.
I notice you just responded to the discussion with an email asking why
I haven't frobulated yet, and I want to chat briefly about what you're
When you understand what Bosslady/Bossman wants, talk about how they want to respond to the email. If they misunderstood the situation and it seems appropriate, you can ask them to send out a follow-up. More often though, you'll end up sending out a short summary something like:
Following up on our offline discussion, I will [move forward with clockwise frobulation/continue to wait for input from Sue/some third thing].
Third: embrace the kabuki. Imagine if Sue says "raterus, stop frobulating!" and then three days later emails your manager "Why did raterus stop frobulating??" Your boss may be irritated enough at Sue to (publicly) ask you to explain (publicly) exactly why you stopped frobulating. Or there may be no complaint, and Boss is just setting you up with a deadline so that you have the leverage to demand that Sue and Bob come to some agreement on frobulation. Or some other goal entirely. The point is, even if it is devious political maneuvering, it may well be meant to support you. Obviously, this goes better if you either know and trust your boss very well, or communicate regularly and clearly with her (preferably both). But talking offline lets you get a sense of this, and respond accordingly.
- If this is part of a pattern of your manager throwing you under the bus, or if you're leaving next week and don't care if everything burns down behind you, the answer might be different. Since you didn't ask about those scenarios, I didn't address them. Even there though, the low-key approach is still often better.
† This can be grating for listeners, but it strongly avoids confrontation. "I have a problem, and I need your help to resolve it." is much harder to react angrily to than "You screwed up and you have to fix it!" Often, it means the same thing, but the framing makes a huge difference‡ .
‡ If you think this is beneath you, I'm sympathetic. Really, I am. Mitigated speech is tiresome and feels dirty. All I can say is: Is this a situation where you want to get weird pissing contests for no good reason, or where you want to solve your actual problems?