I recently replaced the battery and had to have my car jumpstarted. I took it in to see what was drawing the battery down and it was determined that it was the alternator. I had that replaced and saw a puff of smoke, from the solenoid, when I went to start it. I replaced that and now my car tries to start from the battery and bypasses my ignition altogether, without the key even in the ignition.
Sounds like your starter solenoid may be installed incorrectly. The solenoid has several wires going to it. One to the stater, some coming from the battery, and one form the ignition switch. If the wires are connected on the wrong terminals, you can deliver battery voltage to the starter causing it it crank without the key. I suggest seeking help from a mechanic to help you correct the installation of the solenoid and fix the issue you're having to get the car started.
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