Hello, I'm French(I may make grammar mistakes!), I have always had the greatest interest in the way France was governed post Cold War, more specifically the 5eme République. The subject is vast, so I'll mostly focus on the post-war/1970 period.

So first of all, it is important to know that, post WW2, French politics were, to say the least, extremely inefficient and inefective. The 4eme République, whose first government was created in 1946, was the perfect model as to why fully parlimentary systems didn't work in the context at the time. Whilst in other powerful countries at the time, the US, URSS, Britain, a leader with actual power was in place, French presidents at the time held no particular power. The IVeme republique held most of the decision making power to the Assemblée Générale.

As you mentionned, the Communist Party in France post war was the most powerful in Western Europe. A lot of the members of the Resistance during the Occupation were Communist sympathisers, many of whom perished fighting the Nazi occupant. I recall learning in class the Communist party peaked at around 20% of the French population.

But whilst the far left party was strong, the more liberal and right wing political parties weren't weaker, quite the contrary. With the rising civil unrest due to the ineficient leadership of the IVeme République, the current political system was pretty much dommed.

In 1958, Charles de Gaulle, renowned in France for his actions during World War II, essentially leading the French Resistance and Armed Forces post-Capitulation, from 1940 until the end of the war. Managing to hold together France in its darkest hours, during the Occupation, de Gaulle is entrusted the leadership of the country, with the objective to implement a new constitution for the country : the Veme République

The 5th Republique, in which we live in today in France, instored the Suffrage Universel Direct for the election of the President, meaning all eligible citizens for vote will be able to have their voice heard in the elections with their vote. Prior to this decision by de Gaulle, during the previous Republics, a College of "Grands Electeurs", about 50,000 people, local representatives from different areas of France, its colonies and territories, decided who would be the leader of the country. This allows us to further measure the evolution of political parties, more particularly the Communist Party, which we find throughout the years to progressively decrease in importance, the more center-left parties (Parti Socialiste), bringing in over time a lot of voters who identified themselves previously as in favour of the Communist party.

Moreover, the Veme République granted the elected President of France a great deal in powers, executive powers similar to the ones US Presidents or UK Prime-Ministers had. With France finally in the hands of a competent leader who could actually make a difference, the country moved into a great era of progress, the "Trente Glorieuses" the 30 Glorious years.

I consider de Gaulle as to having one of the best minds for Geopolitics and decision-making France has ever had. His government, composed of many people who had participated in the Resistance, finally detained enough executive powers to take action. Thanks to historical support for the government coming from the French population, mostly due to de Gaulle's popularity, many Cold-war related decisions were made:

-In 1958, France was merely a member of NATO. Having not yet followed the US and the UK in the creation of atomic bombs, France had no leverage in the Organisation, and seemed more on the decline in the point of view of the other powers. Charles de Gaulle asked US President Eisenhower to include France in the top leadership of NATO as well as extending NATO treaty coverage to France colonies(Algeria in particular), so as to better insure France's security, and put it in a better place for Command. Eisenhower declined.

-Due to this refusal, de Gaulle saw no other option than to make France more independant from its allies. He progressively withdrawn France from the Commandment of NATO, leading all US Army personnel and equipement to leave France. Charles de Gaulle then took the initiative to start the developement of France's own "dissuasion nucléaire", atomic bombs, which the country officially obtained in 1960.

-Further developement of the C.E.E, Communauté Economique Européenne, the foundations of the European Union. In 1963, along with RFA leader Konrad Adenauer, de Gaulle reinforced greatly the relations between France and Western Germany, a bi-lateral treaty leading to further co-operation between these neighbor nations. This treaty, in my opinion, is the historical marking spot of the final end of Franco-German anemosities, two countries who had fought each other for as long as they existed.

France was later in the 60's drawn into the Algeria War, still seen today by many French as a very controversial set of events, sapped the support for President de Gaulle, and in May 1968, extremely powerful and well supported Demonstrations occured around France. Mostly led by students and young adults with left wing ideals, de Gaulle organised a Référendum, a national scale vote, one the subject of transfering some of the executive powers to more local regions of France. The NO vote won by about 52%, de Gaulle resigned as President.

His former Prime Minister, Georges Pompidou, was elected after him, in 1969.

Now, it's important to know that although de Gaulle lost the vote, he was still renowned by a majority of the French population. To put it simply, many of the French believed it was just time for de Gaulle to stand down and make place for another. A change of leadership could indeed galvanise furthermore the country. As such, while the Parti Socialiste, center-left party gained many and more from the Mai 68 demonstrations, the liberal and right wing parties were still stronger, leading, as sais previously, to Pompidou's election in 1969.

To conclude, while left wing ideologies has always had a presence in post-war France, they never had as much importance as right-wing parties. France's international stance has always been pro-West. France has always proned wide and efficient co-operation with its allies, but it also wanted to keep its complete independance, as well as its pride as a great Nation.