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Use a "better" opamp. The offset is an oldie.
You can do with the 741, however, if you know it's limits.
The offset can be nulled using a trimpot, see the datasheet.
There's nothing you can do about the rise and fall time (edges). The 741 has an output slew rate of 0.5V/µs. This means it takes 2µs to generate a voltage difference of 1V. If your signal is e.g. +-10V, then the OpAmp will take 20V/(0.5V/µs)=40µs toi slew from +10V tp -10V or vice versa.
You will need an OpAmp with a much higher slew rate to overcome this.
Apart from that: using an OpAmp to amplify a square wave by 2 in my opinion is overkill. A simple transistor amplifier can do the same - and much faster.
Show us your circuit diagram.
For calculating a imple OpAmp, there are thousands of web sites. Ask Google, please.
This circuit is inverting. That may be no problem at all for your application. This type of amplifier can sink considerable current when T1 is on. In the off state of T1, the sourceable current is limited by the resistance of R2.
If that is fine for your application, go ahead.
If you need to sink and source current, use a complementary output stage as for example shown here.
???I was thinking about reducing the peak voltage to 6V and amplify it with a gain of 2.
The hfe of a typical transistor is much greater than 10.I need to find a way to get hfe of the transistor to be 10.