Immobiliser sign appears specially in cold morning
 

Hello Everyone,
I have Mazda 3 Hatchback. I am having a issue with starting my car specially in morning or in cold weather. When I turn the Car ON, the immobilizer sign starts to blink on the dashboard. Then I have to play a little bit while then only I can start my car. Even I went to one of the Mazda dealer they changed something else, charged $750 and they said its fixed. And next morning I had same issue and went back to them. But they charged me $180 again for diagnosis and they asked I need to pay $1200 to fix it and even wit this they are not sure if it will be fixed or not.
So, I came here to resolve my issue.
Please help me.

Thank you