One of the problems Elizabeth had when she came to the thrown was cultural development. The idea of the Renaissance had come to Britain in the sixteenth century and people were becoming more interested in ancient ways of life. Not only that, but a new interest and developments in science and machinery (such as the printing press) meant these ideas could be spread quickly.
This was a problem for Elizabeth because she was less able to control what people would think; for example, the belief of a monarchy chosen by God could be questioned if science was to deplete religious beliefs; people would become more educated, learn to think for themselves and there would be more views contradicting the monarchy. This was also a problem when migration followed: new ideas were being bought over, but citizens of England could migrate themselves and leave their new Queen.
Although cultural developments are a problem for Elizabeth, these do not seem to be the most pressing issue at the time.