French lavender (Lavandula stoechas subsp. stoechas) can be tender in the UK so Which? Gardening magazine trialled different methods of overwintering the plants to see which method would work best:

Overwintering outdoors

All our French lavenders survived outside in the mild winter. French lavenders are native to the Med where, although it can get cold in winter, it's drier than the UK and the soils are far more free draining. We therefore added lots of grit to the soil before planting and this probably helped them to survive, especially on the heavy clay of our Glasgow garden. 

Caption: French lavender is more tender than English lavender

Other methods we tried: 

Lifting plants and overwintering indoors

We had thought that lifting our French lavenders (Lavandula stoechas subsp. stoechas) and overwintering them in the greenhouse would work well, as they are often grown in pots. However, all our plants declined in health, looking pale with many dead leaves. They didn't pick up when planted out in late spring and in Yorkshire the difference in foliage colour, compared with the  lavenders overwintered outside, was striking. The lifted plants were a ghostly shade of silver while
the plants grown outside had vivid-green new growth. 

Taking cuttings to overwinter indoors

Both our test sites found it impossible to strike cuttings from these lavenders. We tried semi-ripe cuttings in late summer and might have had more success with softwood cuttings. We tried several times, experimenting with combinations of using hormone rooting powder or none, and various proportions of grit to compost. We made sure to cover the cuttings, but despite our best efforts none rooted.