When you get a flu vaccine do they inject a small amount of the flu virus into you?

Not really. What they do is grow a lot in the lab, harvest the particles and kill it. By injecting the equivalent of dead virus into your body your immune system develops antibodies to the outside shell of the virus. It is not alive and cannot give you the flu.
Yes. A small amount of the virus growth on chick embryos. The virus is killed before injection. The nasal spray vaccine has live attenuated virus.

Related Questions

Does the flu shot actually make you sick? And does the nasal flu virus Mae you sick?

No. The flu shot is a dead virus. That being said your body will react to it and create an immune response to you will create antibodies against the virus. The nasal flu mist is a weakened virus. Again your body will attack it to create an immune response against the virus. If you have a weakened immune system then it is possible that the live nasal virus could make you sick. Read more...
Influenza vaccine. Flu shot side effects may include: slight swelling, redness or pain at vaccination site. If headache, nausea, fever or muscular pain occur they usually resolve within one to two days. You won't get the flu from a flu shot because the virus either contains inactivated virus or doesn't have virus particles in it. Severe side effects are rare. Read more...