There are two separate questions in here:

1. Did Soviet citizens, by and large, support the communist system?
2. Did some number of Soviet citizens support the German invaders during WWII?

To address the second question first, yes, some Soviet citizens did side with the Germans during WWII. However, the vast majority did not. We can look to Ukraine for an example. Over 4.5 million Ukrainians served in the Red Army during WWII, dwarfing the number that served in pro-Nazi and anti-Soviet militias. Ukrainian scholar Peter J. Potichnyj makes the following observation:

In the Soviet army alone were 4.5 million citizens of Ukraine. According to Soviet statistics, 409,668 Ukrainians were awarded medals for bravery in the war; 961 became heroes of the Soviet Union; and 60 per cent of the 250,000-strong Soviet partisan force in Ukraine was Ukrainian.

So in short, while there were pro-Nazi and anti-communist elements among the Soviet population (particularly in Ukraine), the vast majority supported their government at least enough to fight for it against Nazi Germany.

Now, as for the question of whether or not the Soviet people "liked communism," this is a bit harder to quantify. We do have some amount of information; on March 17, 1991, a free referendum (the first and only one in Soviet history) was held, asking a simple yes/no question:

"Do you consider necessary the preservation of the Union of Soviet Socialist Republics as a renewed federation of equal sovereign republics, in which the rights and freedoms of an individual of any nationality will be fully guaranteed?"

A number of republics boycotted the referendum; however, their populations made up only a small fraction of the USSR. Overall, a majority of the population did support maintaining the Union:

The six union republics where separatist aspirations were strongest—Armenia, Estonia, Georgia, Latvia, Lithuania, Moldavia (Moldova)—boycotted the referendum. However, their populations made up only approximately 7 percent of the total USSR population. Overall turnout was 80.0 percent, and 76.4 percent of those participating voted "yes." In Russia, turnout was 75.4 percent, with 71.3 percent voting "yes," while in Ukraine turnout was 83.5 percent, with 70.2 percent voting "yes" (the lowest percentage among all union republics). In all six republics with traditionally Muslim majorities, well over 90 percent voted "yes."

However, can we call this "support for communism," or was it simply a desire to preserve the nation? This is a difficult question to answer. It's fair to assume that most Soviet citizens wanted some kind of reform to their system; however, they certainly didn't want the sort of neoliberal capitalism that they got when the USSR broke up. According to a recent poll, 75% of Russians feel that the Soviet era was the "greatest time" in their country's history. Overall, this is a complex question, with a number of important dimensions.