Can Russians exit Russia on an internal passport?

Not possible. There are only a few countries to which Russians can exit with the internal passport, and Georgia isn’t one of those. Russian citizens can only use the internal passport to visit Belarus, Kazakhstan, Kyrgyzstan, Armenia (a recent addition), the Russian-occupied territories of Georgia that Russia recognizes as independent and Russian-annexed territories of Ukraine.

Possible solutions, other than obtaining a Russian external passport, would involve doing part of the journey by air, such as flying from Russia to Armenia using the internal passport.

There are also complications for entering Russia. You write:

And obviously entering Russia on an internal passport shouldn’t be a problem.

That’s not so obvious at all. The law that governs entering and leaving Russia is the law “О порядке выезда из Российской Федерации и въезда в Российскую Федерацию”. Article 6 of the law states:

Выезд из Российской Федерации и въезд в Российскую Федерацию граждане Российской Федерации осуществляют по действительным документам, удостоверяющим личность гражданина Российской Федерации за пределами территории Российской Федерации.

It says, briefly, that Russian citizens have to use the external passport both for entering and leaving the country. In practice, this should not mean that entry will be denied, but the usual situation would be that the person gets detained at the border until the authorities are convinced they are who they claim to be, then the person gets fined and allowed into the country. If your friend also happens to be a resident of Sweden, and not Russia, it might further delay the process of being allowed in on an internal passport.

Yes, you can’t exit the Russia with internal passport on border points with countries which have no visa-free agreement with Russia. More over, there are very few countries which do accept Russians without international passport.

Even if you don’t need a visa, you need a valid passport, because Russian internal passport is intended to prove your identity within Russian borders, and not outside them (it’s a Federal Law about Russian internal passport, link in Russian)

So, basically, your friend can’t prove her identity with internal Russian passport outside Russia. Moreover, the absence of it may lead to problems with entering the country, as, strictly speaking, you’re still outside the Russia, so you need the international passport.

Another update:
But! The Federal Law about rules for enter/exit the Russia says that no Russian citizens can be denied from entrance the Russia. In case of losing the passport outside the Russia one should contact local Russian embassies/consulates for getting the document proving one’s identity during entrance the Russia.

In another words, your trip is still an option, but may lead you to numerous checks on entrance the Russia, even if you’ll manage out to get the temporary identity document. This is an unusual case for border officers, so you may lose a lot of time.

I suggest your friend to contact the Russian Consulate in Sweden and ask them about this problem. Maybe they will provide an easy way to resolve this (like providing her the temporary document), but it looks like your friend had to get her international passport, either from Sweden or from Russia. It’s about the protocol – she needs a legal id proving that she can enter and exit the Russia, no matter where she goes from/to.

Update about the international passports: Right now there are two types of IP for Russians being available. First type is so called “old template” (иностранный паспорт старого образца), which is still available to get, and it’s validity is 5 years from issue date. Second type is so called “new template” (иностранный паспорт нового образца) or “biometrics passport” (биометрический паспорт), which validity is 10 years.

What are the differences?

1. Old is cheaper
2. Old doesn’t contain biometric information, and has a simple photo in it, so for a new one Embassy need a technology update.
3. Old is valid two times less comparing to old one.
4. Old passport can be issued for a child under 12 years without bringing the child in Embassy, and can be done in two weeks.
5. For old one you need photos, for new one you don’t.

So I assume that the old one was the easiest option, that’s why the passports were valid only for a 5 years.

All in all, I strongly suggest your friend to contact the Embassy for a passport renewal, and only after that do the trip. This is safest and easiest way to resolve this problem. She can do that in Consulate General of the Russian Federation in Gothenburg (Russian link). It will be done in up to 3 months, price is 276 SEK for old template, and 736 SEK for a new one.