First of all, you may be invited based on professional profile. Do not accept an invitation for a review outside your expertise field.

Second, you need to mention such invitation to your teacher, especially if you are working on it in day-time at your own institute.

If you feel an expert and you decide to work in the evening, outside your normal working hours and this review does not influence your day-time acitivities or ethical profile, than you could accept the invitation. Still it would be wise to mention it to your teacher. Maybe he is interested to join forces.

Authorship should be based on contribution to the paper not on work groups or labs. However undertaking a review can be a challenging task and so you might want to work with a co-author or two to develop the paper. There are a number of useful guides on journal paper authorship (eg http://www.icmje.org/recommendations/browse/roles-and-responsibilities/defining-the-role-of-authors-and-contributors.html). Most journals will also state their standards and a link to specific guidelines (or you can contact the editor).

I would suggest you discuss with your supervisor as they may have useful thoughts on how to go about it and may offer to read and review the paper prior to submission if they are not authoring it. You never know, there may be substantive contributions they (or someone else in the lab) could make to the paper that would warrant co-authorship.

Good luck with your writing.

First of all, you may be invited based on professional profile. Do not accept an invitation for a review outside your expertise field.

Second, you need to mention such invitation to your teacher, especially if you are working on it in day-time at your own institute.

If you feel an expert and you decide to work in the evening, outside your normal working hours and this review does not influence your day-time acitivities or ethical profile, than you could accept the invitation. Still it would be wise to mention it to your teacher. Maybe he is interested to join forces.

If you are being asked to review research where the author is one of your supervisors, this is indeed an ethical issue and I would advise informing the editor of the journal about this conflict of interest.