I analysed and tried 4. Nxe4 h6 a little.

Key idea: It's for a Bf5-player that after Ng3 the bishop can retreat to h7 in one move instead of two (please note that for white this is not clear: You may be an Nbd7-player trying to avoid Ng5-lines). So a line which illustrates the idea would be 1. e4 c6  2. d4 d5  3. Nc3 dxe4  4. Nxe4 h6  5. Nf3 Bf5  6. Ng3 Bh7. Now the typical h2-h4-h5 does not make much sense for white: He would be one tempo down compared to the main lines. Also the dangerous lines with Nh3-Nf4 (or via N1e2) loose some of their bite.

Drawback: If White chooses a line without h2-h4-h5 in which blacks h6 is not very useful black is a tempo down. After 4. Nxe4 Bf5 these lines are with the exception of the Nh3-Nf4 lines relatively harmless, but with a tempo more they get stronger. Example: 4. Nxe4 h6  5. Lc4 Bf5  6. Ng3 Bh7  and now white can go for f4-plans either directly (7. f4 e6  8. Nf3 [or even immediatelly f5] Bd6 as in Sorokin - Deviatkin 2006) or later (7. N1e2 Nf6 8. 0-0 e6  9. f4 as in Nastase - Lupulescu 2015 but instead of blacks h5 I would prefer maybe Ndb7 10. f5 e5).

Specialities: White can try to be clever and play after h6 5. Ng3 which prevents Bf5. It's not bad to answer that with the funny h5. Another special idea for White is 5. Bd3 because it is very risky to take on d4 and to fall back further in development. After 5. Bd3 Bf5  white can go for setups in which he keeps the Ne4 in the center e.g. with N1e2 and f3 or play 6. Qf3. Should be all ok for black but the patterns are somehow different to usual Bf5-CaroKanns.

Conclusion: I like the h6-line and think it is not worse than the main lines with Bf5, Nd7 or Nf6, but hey: I'm not rated 2700 so what does it mean. In the Bf5-system the h4-lines get weaker or not meaningful and the lines without h4 a little stronger.

Please note that you can also find info about this line on the net under the name Burmakin-variation named after the Caro-Kann expert Vladimir Burmakin who played it from time to time.

Happy to discuss this further and learn more ...