'Objective' is open to many interpretations. For an objective morality I should mark out four positions (which isn't to say there aren't others).
This is the view that moral judgements can be true or false; that some are true; and that some are known to be true. The ontology behind this requires really existing moral properties or facts. A moral epistemology is needed to explain how we can become aware of the moral properties or facts by virtue of which we know that some moral judgements are true.
Ethics of divine commands
There is no single, uncontestable statement of this position but it can be roughly stated as follows : divine commands have moral force in and of themselves - in which case at least some (perhaps all) moral obligations derive from divine commands.
It is possible to see an ethics of divine commands as a form of moral realism but moral realism can be held without commitment to an ethics of divine commands. This is why I have separated them.
Finally, at least in this brief survey, morality can be seen as a form of rationality. This is clear in Kant, whose view is that it is irrational to act on maxims, principles of conduct, if it is logically impossible for everyone to act on such maxims or principles. (Kant's position is more complex but this is at its heart.)
▻ Kant and truth
Kant's moral philosophy makes no use whatever of the concept of truth. The objectivity of morality rests on the sole basis that rational agents cannot consistently reject moral requirements. Such requirements derive their full and exclusive force from their rationality. (Brian K. Powell, 'Kant and Kantians on "The Normative Question", Ethical Theory and Moral Practice, Vol. 9, No. 5 (Nov., 2006), pp. 535-544 : 535.) More specifically, rationality requires me to act on maxims or principles on which it is logically possible for everyone to act.
For instance, suppose I have been invited to a 'bring a bottle party' and I decide to take cheap and drink expensive. My maxim is (Kantianly construed) : 'Whenever one is invited to a bring a bottle party, take cheap wine and drink the expensive wine that others have brought'. It is logically impossible for this maxim to be universalised : if everyone took cheap wine in order to drink the expensive wine brought by others, there would be no expensive wine for anyone to drink.
These are the lines along which Kant's ethical theory runs. They have nothing to do with truth, everything to do with consistent universalisability.
This embodies a form of instrumental rationality. An intrinsic good or range of intrinsic goods, or a a set of intrinsic valuings, is identified. These have in themselves no moral character. Morality comes into play when as a social institution it is given the task of maximising the occurrence of these goods or valuings through the consequences of actions. An action is right if, for instance, it maximises through its consequences the occurrence of such intrinsic goods or valuings as pleasure, health, knowledge or whatever the list of intrinsic goods or valuings comprises. Utilitarian moral judgements are not true, or taken by utilitarians to be true; their moral commendation is their instrumental rationality in the maximising of intrinsic goods or valuings.
G. Sayre-McCord, Essays on Moral Realism (Cornell Paperbacks), SBN 10: 0801495415 / ISBN 13: 9780801495410
Published by Cornell University Press, 1988.
Michael J. Harris, Divine Command Ethics, ISBN 10: 1138869767 / ISBN 13: 9781138869769
Published by Routledge, 2015.
D. A. Rees, 'The Ethics of Divine Commands', Proceedings of the Aristotelian Society, New Series, Vol. 57 (1956 - 1957), pp. 83-
Onora O'Neill, Constructions of Reason: Explorations of Kant's Practical Philosophy, ISBN 10: 0521388163 / ISBN 13: 9780521388160
Published by Cambridge University Press, 2016.
Otfried Hoffe, Immanuel Kant (S U N Y Series in Ethical Theory), ISBN 10: 0791420949 / ISBN 13: 9780791420942
Published by State University of New York Press, 1994.
Brian K. Powell, 'Kant and Kantians on "The Normative Question", Ethical Theory and Moral Practice, Vol. 9, No. 5 (Nov., 2006), pp. 535-544.
Geoffrey Thomas, In Introduction to Ethics, ISBN 10: 0715624318 / ISBN 13: 9780715624319
Published by Gerald Duckworth & Co Ltd, 1997.
J. J. C. Smart, Bernard Williams, Utilitarianism: For and Against, ISBN 10: 0521202973 / ISBN 13: 9780521202978
Published by Cambridge University Press, 1973.